💡Okay, so I have this problem about a parabola, and I need to solve two parts. Let me try to understand what's given and what I need to find.First, the parabola is given by the equation ( C: x^2 = 2py ) where ( p > 0 ). There's a point ( P(2, y_0) ) that lies on this parabola and is at a distance of 2 from the focus ( F ). For part (1), I need to find the equation of the parabola ( C ) and its directrix equation. Alright, let's recall some properties of parabolas. The standard form ( x^2 = 4py ) has its vertex at the origin, opens upwards, focus at ( (0, p) ), and directrix at ( y = -p ). Wait, but in our case, the equation is ( x^2 = 2py ). So comparing to the standard form, ( 4p' = 2p ), so ( p' = p/2 ). So the focus should be at ( (0, p/2) ) and the directrix at ( y = -p/2 ).Given that point ( P(2, y_0) ) lies on the parabola, so it must satisfy the equation ( 2^2 = 2p y_0 ), which simplifies to ( 4 = 2p y_0 ), so ( y_0 = 2/p ).Also, the distance from ( P ) to the focus ( F(0, p/2) ) is 2. So, using the distance formula, the distance between ( (2, y_0) ) and ( (0, p/2) ) is 2.So, let's write that equation:( sqrt{(2 - 0)^2 + (y_0 - p/2)^2} = 2 )Squaring both sides:( 4 + (y_0 - p/2)^2 = 4 )Subtract 4 from both sides:( (y_0 - p/2)^2 = 0 )So, ( y_0 - p/2 = 0 ), which implies ( y_0 = p/2 )But earlier, we found that ( y_0 = 2/p ). So, setting these equal:( p/2 = 2/p )Multiply both sides by ( p ):( p^2 / 2 = 2 )Multiply both sides by 2:( p^2 = 4 )Since ( p > 0 ), ( p = 2 )So, the equation of the parabola is ( x^2 = 2 * 2 * y = 4y ). So, ( x^2 = 4y ).The directrix is ( y = -p/2 = -2/2 = -1 ). So, directrix is ( y = -1 ).Okay, that seems solid. So part (1) is done.Now, moving on to part (2). Given point ( M(2, -1) ), the line ( y = kx + 1 ) (where ( k neq 0 )) intersects the parabola ( C ) at points ( A ) and ( B ). The slopes of lines ( MA ) and ( MB ) are ( k_1 ) and ( k_2 ) respectively. We need to find ( frac{1}{k_1} + frac{1}{k_2} ).Hmm, okay. So, first, let's note that the parabola is ( x^2 = 4y ). The line is ( y = kx + 1 ). Let's find the points of intersection ( A ) and ( B ).To find the points of intersection, substitute ( y = kx + 1 ) into ( x^2 = 4y ):( x^2 = 4(kx + 1) )Simplify:( x^2 = 4kx + 4 )Bring all terms to one side:( x^2 - 4kx - 4 = 0 )This is a quadratic in ( x ). Let me denote the roots as ( x_1 ) and ( x_2 ). So, the solutions are:( x = [4k pm sqrt{(4k)^2 + 16}]/2 )But actually, I might not need the exact roots. Instead, since it's a quadratic, I can use Vieta's formulas.For quadratic ( ax^2 + bx + c = 0 ), sum of roots ( x_1 + x_2 = -b/a ), product ( x_1 x_2 = c/a ).Here, ( a = 1 ), ( b = -4k ), ( c = -4 ). So:( x_1 + x_2 = 4k )( x_1 x_2 = -4 )Okay, so points ( A ) and ( B ) have coordinates ( (x_1, y_1) ) and ( (x_2, y_2) ), where ( y_1 = kx_1 + 1 ) and ( y_2 = kx_2 + 1 ).Now, we need to find the slopes ( k_1 ) and ( k_2 ) of lines ( MA ) and ( MB ).Point ( M ) is ( (2, -1) ). So, the slope ( k_1 ) is:( k_1 = frac{y_1 - (-1)}{x_1 - 2} = frac{y_1 + 1}{x_1 - 2} )Similarly, ( k_2 = frac{y_2 + 1}{x_2 - 2} )So, ( frac{1}{k_1} = frac{x_1 - 2}{y_1 + 1} ) and ( frac{1}{k_2} = frac{x_2 - 2}{y_2 + 1} )Therefore, ( frac{1}{k_1} + frac{1}{k_2} = frac{x_1 - 2}{y_1 + 1} + frac{x_2 - 2}{y_2 + 1} )But ( y_1 = kx_1 + 1 ), so ( y_1 + 1 = kx_1 + 2 ). Similarly, ( y_2 + 1 = kx_2 + 2 ).So, substituting back:( frac{1}{k_1} + frac{1}{k_2} = frac{x_1 - 2}{kx_1 + 2} + frac{x_2 - 2}{kx_2 + 2} )Hmm, this looks a bit complicated. Maybe I can combine these fractions.Let me write it as:( frac{(x_1 - 2)(kx_2 + 2) + (x_2 - 2)(kx_1 + 2)}{(kx_1 + 2)(kx_2 + 2)} )Let me expand the numerator:First term: ( (x_1 - 2)(kx_2 + 2) = kx_1x_2 + 2x_1 - 2kx_2 - 4 )Second term: ( (x_2 - 2)(kx_1 + 2) = kx_1x_2 + 2x_2 - 2kx_1 - 4 )Adding these together:( kx_1x_2 + 2x_1 - 2kx_2 - 4 + kx_1x_2 + 2x_2 - 2kx_1 - 4 )Combine like terms:- ( kx_1x_2 + kx_1x_2 = 2k x_1x_2 )- ( 2x_1 + 2x_2 = 2(x_1 + x_2) )- ( -2kx_2 - 2kx_1 = -2k(x_1 + x_2) )- ( -4 - 4 = -8 )So numerator becomes:( 2k x_1x_2 + 2(x_1 + x_2) - 2k(x_1 + x_2) - 8 )Factor terms:( 2k x_1x_2 + 2(1 - k)(x_1 + x_2) - 8 )Now, let's substitute the known values from Vieta's formulas:( x_1 + x_2 = 4k )( x_1x_2 = -4 )So, substituting:( 2k*(-4) + 2(1 - k)*(4k) - 8 )Calculate each term:- ( 2k*(-4) = -8k )- ( 2(1 - k)*(4k) = 8k(1 - k) = 8k - 8k^2 )- The constant term is -8So, numerator becomes:( -8k + 8k - 8k^2 - 8 )Simplify:- ( -8k + 8k = 0 )- So, numerator is ( -8k^2 - 8 )Now, let's compute the denominator:( (kx_1 + 2)(kx_2 + 2) )Expand this:( k^2x_1x_2 + 2kx_1 + 2kx_2 + 4 )Again, substitute ( x_1 + x_2 = 4k ) and ( x_1x_2 = -4 ):( k^2*(-4) + 2k*(4k) + 4 )Calculate each term:- ( k^2*(-4) = -4k^2 )- ( 2k*(4k) = 8k^2 )- The constant term is 4So, denominator becomes:( -4k^2 + 8k^2 + 4 = 4k^2 + 4 )So, now, putting numerator and denominator together:( frac{-8k^2 - 8}{4k^2 + 4} )Factor numerator and denominator:Numerator: ( -8(k^2 + 1) )Denominator: ( 4(k^2 + 1) )So, ( frac{-8(k^2 + 1)}{4(k^2 + 1)} = frac{-8}{4} = -2 )Wow, that simplified nicely! So, ( frac{1}{k_1} + frac{1}{k_2} = -2 )Let me just verify if I made any mistakes in the algebra.Starting from the numerator:( 2k x_1x_2 + 2(1 - k)(x_1 + x_2) - 8 )Plugging in ( x_1x_2 = -4 ) and ( x_1 + x_2 = 4k ):( 2k*(-4) + 2(1 - k)*(4k) - 8 = -8k + 8k - 8k^2 - 8 = -8k^2 - 8 ) Correct.Denominator:( k^2x_1x_2 + 2k(x_1 + x_2) + 4 = -4k^2 + 8k^2 + 4 = 4k^2 + 4 ) Correct.So, the ratio is ( (-8k^2 -8)/(4k^2 +4) = -2 ). So, yes, that's correct.Therefore, the value is -2.**Final Answer**(1) The equation of the parabola is (boxed{x^2 = 4y}) and its directrix is (boxed{y = -1}).(2) The value of (frac{1}{k_1} + frac{1}{k_2}) is (boxed{-2}).

💡Alright, so I've got this problem here about a line and a circle intersecting, and I need to find the sum of the distances from a specific point P to the two intersection points A and B. Let me try to break this down step by step.First, let me understand the given information. There's a line l given in parametric form:x = 3 - (√2 / 2) * ty = √5 + (√2 / 2) * twhere t is the parameter. Then, there's a circle C given in polar coordinates:ρ = 2√5 sinθAnd the point P is at (3, √5). I need to find |PA| + |PB|, where A and B are the points where line l intersects circle C.Okay, so I think the first step is to convert the polar equation of the circle into rectangular (Cartesian) coordinates because the line is already given in parametric form, which is easier to work with in Cartesian coordinates.The polar equation is ρ = 2√5 sinθ. I remember that in polar coordinates, ρ = 2a sinθ represents a circle with radius a, centered at (0, a) in Cartesian coordinates. So, in this case, a would be √5, meaning the circle has a radius of √5 and is centered at (0, √5). Let me verify that.Yes, the general form is ρ = 2a sinθ, which translates to x² + (y - a)² = a² in Cartesian coordinates. So plugging in a = √5, the equation becomes:x² + (y - √5)² = (√5)²Simplifying that, it becomes:x² + y² - 2√5 y + 5 = 5Wait, subtracting 5 from both sides gives:x² + y² - 2√5 y = 0So that's the Cartesian equation of circle C. Got it.Now, the parametric equations of line l are:x = 3 - (√2 / 2) ty = √5 + (√2 / 2) tI need to find where this line intersects the circle. So I can substitute the expressions for x and y from the parametric equations into the circle's equation.Let me write that out:x² + y² - 2√5 y = 0Substituting x and y:[3 - (√2 / 2) t]² + [√5 + (√2 / 2) t]² - 2√5 [√5 + (√2 / 2) t] = 0Okay, that looks a bit messy, but let's expand each term step by step.First, expand [3 - (√2 / 2) t]²:= 3² - 2 * 3 * (√2 / 2) t + (√2 / 2 t)²= 9 - 3√2 t + (2 / 4) t²= 9 - 3√2 t + (1/2) t²Next, expand [√5 + (√2 / 2) t]²:= (√5)² + 2 * √5 * (√2 / 2) t + (√2 / 2 t)²= 5 + √10 t + (2 / 4) t²= 5 + √10 t + (1/2) t²Now, expand the third term: -2√5 [√5 + (√2 / 2) t]= -2√5 * √5 - 2√5 * (√2 / 2) t= -2 * 5 - √10 t= -10 - √10 tNow, putting all these together into the equation:[9 - 3√2 t + (1/2) t²] + [5 + √10 t + (1/2) t²] + [-10 - √10 t] = 0Let me combine like terms step by step.First, the constants:9 + 5 - 10 = 4Next, the terms with t:-3√2 t + √10 t - √10 tWait, the √10 t and -√10 t cancel each other out, so we're left with -3√2 t.Now, the t² terms:(1/2) t² + (1/2) t² = t²So putting it all together:t² - 3√2 t + 4 = 0Hmm, so the quadratic equation in terms of t is:t² - 3√2 t + 4 = 0Let me write that down:t² - 3√2 t + 4 = 0I need to solve for t. Let me use the quadratic formula:t = [3√2 ± √( (3√2)^2 - 4 * 1 * 4 ) ] / 2First, compute the discriminant:(3√2)^2 = 9 * 2 = 184 * 1 * 4 = 16So discriminant is 18 - 16 = 2So,t = [3√2 ± √2] / 2Therefore, the two solutions are:t1 = [3√2 + √2] / 2 = (4√2) / 2 = 2√2t2 = [3√2 - √2] / 2 = (2√2) / 2 = √2So, t1 = 2√2 and t2 = √2Therefore, the points A and B correspond to t = 2√2 and t = √2.Now, I need to find the coordinates of points A and B.Using the parametric equations:For t = 2√2:x = 3 - (√2 / 2) * 2√2 = 3 - (√2 * 2√2) / 2 = 3 - (2 * 2) / 2 = 3 - 4 / 2 = 3 - 2 = 1y = √5 + (√2 / 2) * 2√2 = √5 + (√2 * 2√2) / 2 = √5 + (2 * 2) / 2 = √5 + 4 / 2 = √5 + 2So point A is (1, √5 + 2)For t = √2:x = 3 - (√2 / 2) * √2 = 3 - ( (√2 * √2) / 2 ) = 3 - (2 / 2) = 3 - 1 = 2y = √5 + (√2 / 2) * √2 = √5 + ( (√2 * √2) / 2 ) = √5 + (2 / 2) = √5 + 1So point B is (2, √5 + 1)Now, point P is given as (3, √5). I need to find |PA| + |PB|.First, let's find |PA|, the distance between P(3, √5) and A(1, √5 + 2).Using the distance formula:|PA| = √[ (3 - 1)^2 + (√5 - (√5 + 2))^2 ]Simplify:= √[ (2)^2 + (-2)^2 ]= √[4 + 4]= √8= 2√2Next, find |PB|, the distance between P(3, √5) and B(2, √5 + 1).Again, using the distance formula:|PB| = √[ (3 - 2)^2 + (√5 - (√5 + 1))^2 ]Simplify:= √[ (1)^2 + (-1)^2 ]= √[1 + 1]= √2So, |PA| = 2√2 and |PB| = √2Therefore, |PA| + |PB| = 2√2 + √2 = 3√2Wait, that seems straightforward, but let me double-check my calculations to be sure.First, for point A:x = 1, y = √5 + 2Distance from P(3, √5):Δx = 3 - 1 = 2Δy = √5 - (√5 + 2) = -2Distance squared: 2² + (-2)² = 4 + 4 = 8Square root of 8 is 2√2. Correct.For point B:x = 2, y = √5 + 1Distance from P(3, √5):Δx = 3 - 2 = 1Δy = √5 - (√5 + 1) = -1Distance squared: 1² + (-1)² = 1 + 1 = 2Square root of 2 is √2. Correct.So, adding them together: 2√2 + √2 = 3√2Hmm, that seems correct. But wait, I recall that in parametric equations, the parameter t often relates to the distance from a particular point, especially if the parametric equations are set up that way.Looking back at the parametric equations of line l:x = 3 - (√2 / 2) ty = √5 + (√2 / 2) tI notice that the coefficients of t in both x and y are ±√2 / 2. Let me check if this is a unit vector.The direction vector of the line is (-√2 / 2, √2 / 2). Let's compute its magnitude:√[ (-√2 / 2)^2 + (√2 / 2)^2 ] = √[ (2 / 4) + (2 / 4) ] = √[4 / 4] = √1 = 1Yes, it's a unit vector. That means that the parameter t in the parametric equations represents the actual distance from the point (3, √5) along the line l. So, when t = 0, we are at point P(3, √5). As t increases, we move along the line in one direction, and as t decreases, we move in the opposite direction.Therefore, the parameter t directly corresponds to the distance from P. So, if the solutions for t are t1 = 2√2 and t2 = √2, then |PA| and |PB| are simply |t1| and |t2|, respectively.Since both t1 and t2 are positive, |PA| + |PB| = t1 + t2 = 2√2 + √2 = 3√2.This matches my earlier calculation. So, that's consistent.Alternatively, I could have used the property that for a quadratic equation at² + bt + c = 0, the sum of the roots is -b/a. In our case, the equation was t² - 3√2 t + 4 = 0, so the sum of the roots t1 + t2 = 3√2. Since both roots are positive, the sum |PA| + |PB| is just 3√2.So, whether I compute the distances directly or use the properties of the quadratic equation, I arrive at the same answer.Therefore, I'm confident that |PA| + |PB| is 3√2.**Final Answer**The value of (|PA| + |PB|) is (boxed{3sqrt{2}}).

💡First, determine the number of pizza slices left after James's friend eats. James ordered an 8-piece pizza, and his friend ate 2 slices. So, there are 8 - 2 = 6 slices remaining.Next, James eats 3 slices out of the remaining 6 slices.To find the ratio of the number of slices James eats to the number of slices left after his friend eats, set up the ratio as 3 (slices James eats) to 6 (slices left after friend eats).Finally, simplify the ratio by dividing both numbers by their greatest common divisor, which is 3. This results in a simplified ratio of 1 to 2.

💡Okay, so I need to figure out how many different possible values result when I take two positive odd integers less than 15, multiply them together, subtract their sum, and then see how many unique results I can get. Hmm, let me break this down step by step.First, let me list out all the positive odd integers less than 15. That should be straightforward. Starting from 1, the odd numbers are 1, 3, 5, 7, 9, 11, and 13. So, there are 7 numbers in total. Since the problem says "not necessarily distinct," that means I can use the same number twice if I want. So, I can have pairs like (1,1), (1,3), (3,3), and so on.Now, the expression we're dealing with is the product of the two numbers minus their sum. Let me write that down: if the two numbers are p and q, then the expression is pq - (p + q). Hmm, maybe I can simplify this expression a bit to make it easier to work with. Let me try factoring or rearranging it.Looking at pq - p - q, I notice that I can factor this expression. Let me see: pq - p - q is the same as (p - 1)(q - 1) - 1. Wait, how does that work? Let me check:(p - 1)(q - 1) = pq - p - q + 1. So, if I subtract 1 more, I get pq - p - q + 1 - 1 = pq - p - q. Yes, that's correct. So, pq - p - q is equal to (p - 1)(q - 1) - 1. That seems useful because now I can think about the product of (p - 1) and (q - 1) and then subtract 1.So, let me define new variables to make this clearer. Let me set a = p - 1 and b = q - 1. Then, the expression becomes ab - 1. Now, since p and q are odd numbers, subtracting 1 from each will make them even numbers. Let me list out the possible values of a and b.Original p and q: 1, 3, 5, 7, 9, 11, 13.So, a = p - 1: 0, 2, 4, 6, 8, 10, 12.Similarly, b = q - 1: 0, 2, 4, 6, 8, 10, 12.So, now I have a and b each ranging over the even numbers from 0 to 12. The expression ab - 1 will give me the value I'm looking for. My task is to find how many distinct values ab - 1 can take as a and b range over these even numbers.Since a and b can each be 0, 2, 4, 6, 8, 10, or 12, I need to compute all possible products ab and then subtract 1 from each. But instead of computing all 49 possible products (since there are 7 choices for a and 7 for b), maybe I can find a smarter way to count the distinct results.First, let me note that ab can range from 0*0=0 up to 12*12=144. So, ab - 1 can range from -1 up to 143. But of course, not all numbers in this range will be achievable because a and b are restricted to even numbers.Wait, actually, since a and b are even, their product ab will also be even (since even times even is even). Therefore, ab is even, so ab - 1 will be odd. That means all the results will be odd numbers from -1 up to 143. So, the possible results are all odd integers in that range. But not all of them will necessarily be achievable because ab might skip some numbers.So, my next thought is to figure out how many distinct odd numbers can be expressed as ab - 1 where a and b are even numbers from 0 to 12. Alternatively, since ab is even, ab - 1 is odd, so we can think of ab as (result + 1), which must be even. Therefore, result + 1 must be equal to ab, where a and b are even numbers from 0 to 12.So, another way to phrase the problem is: how many distinct even numbers can be expressed as ab where a and b are even numbers from 0 to 12? Then, subtracting 1 from each of those will give us the distinct results we're looking for.Therefore, if I can find the number of distinct even products ab, then the number of distinct results will be the same as the number of distinct even products, since subtracting 1 is a bijection (one-to-one correspondence) between the set of even numbers and the set of odd numbers.So, now, my problem reduces to finding how many distinct even numbers can be formed by multiplying two even numbers a and b where a and b are in {0, 2, 4, 6, 8, 10, 12}.Let me list all possible products ab:First, let me note that 0 multiplied by anything is 0. So, if either a or b is 0, the product ab is 0. So, 0 is definitely one of the products.Now, let's consider the non-zero products. Since a and b are even numbers from 2 to 12, let's list them:a: 2, 4, 6, 8, 10, 12b: 2, 4, 6, 8, 10, 12So, the products will be:2*2=42*4=82*6=122*8=162*10=202*12=244*2=84*4=164*6=244*8=324*10=404*12=486*2=126*4=246*6=366*8=486*10=606*12=728*2=168*4=328*6=488*8=648*10=808*12=9610*2=2010*4=4010*6=6010*8=8010*10=10010*12=12012*2=2412*4=4812*6=7212*8=9612*10=12012*12=144Now, let me list all these products and see which ones are unique:From the above, the products are:4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, 100, 120, 144.Wait, let me check each product:- 4: appears once- 8: appears twice (2*4 and 4*2)- 12: appears twice (2*6 and 6*2)- 16: appears three times (2*8, 4*4, 8*2)- 20: appears twice (2*10 and 10*2)- 24: appears four times (2*12, 4*6, 6*4, 12*2)- 32: appears twice (4*8 and 8*4)- 36: appears once (6*6)- 40: appears twice (4*10 and 10*4)- 48: appears four times (4*12, 6*8, 8*6, 12*4)- 60: appears twice (6*10 and 10*6)- 64: appears once (8*8)- 72: appears twice (6*12 and 12*6)- 80: appears twice (8*10 and 10*8)- 96: appears twice (8*12 and 12*8)- 100: appears once (10*10)- 120: appears twice (10*12 and 12*10)- 144: appears once (12*12)So, the distinct products are:4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, 100, 120, 144.That's 18 distinct products. But wait, we also have the product 0, which occurs when either a or b is 0. So, including 0, we have 19 distinct even products.But wait, let me double-check. When a or b is 0, the product is 0. So, ab can be 0 or any of the above 18 products. So, total distinct even products are 19.But hold on, in the original problem, p and q are positive odd integers less than 15. So, p and q can be 1, which would make a = p - 1 = 0. So, 0 is a valid value for a or b, which means ab can indeed be 0. So, 0 is included.Therefore, the total number of distinct even products ab is 19. Therefore, the number of distinct results ab - 1 is also 19, since subtracting 1 from each even product gives a unique odd number.Wait, but earlier I thought the answer was 28. Hmm, maybe I made a mistake in my initial reasoning.Wait, let me go back. I think I might have confused the number of distinct products with the number of distinct results after subtracting 1. Let me clarify.If ab can take 19 distinct even values (including 0), then ab - 1 will take 19 distinct odd values. However, when ab = 0, ab - 1 = -1. So, the results will be -1, 3, 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 135, 143.Wait, let me list them:ab: 0, 4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, 100, 120, 144.So, ab - 1: -1, 3, 7, 11, 15, 19, 23, 31, 35, 39, 47, 59, 63, 71, 79, 95, 99, 119, 143.Wait, that's 19 distinct values. So, why did the initial thought say 28? Maybe I made a mistake in the initial reasoning.Wait, perhaps I didn't consider that some products might result in the same ab - 1 value. Let me check.Wait, no, each ab is unique, so ab - 1 should also be unique. Because if ab1 ≠ ab2, then ab1 - 1 ≠ ab2 - 1. So, the number of distinct ab - 1 should be equal to the number of distinct ab.But in my earlier list, I have 19 distinct ab values, so ab - 1 should also have 19 distinct values. However, the initial answer was 28, which suggests I might have missed something.Wait, maybe I made a mistake in counting the distinct ab values. Let me recount the distinct ab values.From the products above, the distinct ab values are:0, 4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, 100, 120, 144.That's 19 distinct values. So, ab - 1 would be 19 distinct values.But wait, in the initial problem, the user said the answer was 28. So, perhaps my approach is incorrect.Wait, let me think again. Maybe I should consider that p and q are positive odd integers less than 15, so p and q can be 1, 3, 5, 7, 9, 11, 13. So, a = p - 1 can be 0, 2, 4, 6, 8, 10, 12. Similarly for b.But when I compute ab - 1, I have to consider all possible pairs (a, b), which are 7x7=49 pairs. But many of these will result in the same ab - 1 value.Wait, but earlier I thought that the number of distinct ab is 19, so ab - 1 is 19. But maybe I'm missing some products because when a or b is 0, ab is 0, but when both are non-zero, ab can be other even numbers.Wait, let me try a different approach. Let me list all possible ab - 1 values.Since a and b can be 0, 2, 4, 6, 8, 10, 12, let's compute ab for each pair and then subtract 1.But that's 49 computations, which is time-consuming, but maybe necessary to ensure accuracy.Let me create a table:a b | 0 2 4 6 8 10 12-----------------------------------------0 | 0 0 0 0 0 0 02 | 0 4 8 12 16 20 244 | 0 8 16 24 32 40 486 | 0 12 24 36 48 60 728 | 0 16 32 48 64 80 9610 | 0 20 40 60 80 100 12012 | 0 24 48 72 96 120 144Now, let's list all the ab values:From the table:Row 0: 0,0,0,0,0,0,0Row 2: 0,4,8,12,16,20,24Row 4: 0,8,16,24,32,40,48Row 6: 0,12,24,36,48,60,72Row 8: 0,16,32,48,64,80,96Row 10: 0,20,40,60,80,100,120Row 12: 0,24,48,72,96,120,144Now, let's collect all these ab values and count the distinct ones.Starting from the smallest:0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144.Wait, that's 19 distinct ab values. So, ab - 1 would be:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 distinct values.But wait, in the initial problem, the user said the answer was 28. So, I must have made a mistake somewhere.Wait, perhaps I didn't consider that when a or b is 0, ab - 1 is -1, but maybe there are other ways to get the same result. Wait, no, because ab is unique for each pair, so ab - 1 should also be unique.Wait, but in the table, ab can be 0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144. So, 19 distinct ab values, leading to 19 distinct ab - 1 values.But the initial answer was 28, so I must have missed something.Wait, maybe I made a mistake in the initial transformation. Let me double-check.The original expression is pq - p - q. I rewrote it as (p - 1)(q - 1) - 1. Let me verify:(p - 1)(q - 1) = pq - p - q + 1. So, (p - 1)(q - 1) - 1 = pq - p - q + 1 - 1 = pq - p - q. Yes, that's correct.So, the transformation is correct. Therefore, the number of distinct values of pq - p - q is equal to the number of distinct values of (p - 1)(q - 1) - 1, which is the same as the number of distinct ab - 1 where a and b are even numbers from 0 to 12.But according to my table, there are 19 distinct ab values, leading to 19 distinct results. So, why does the initial answer say 28?Wait, perhaps I made a mistake in counting the distinct ab values. Let me recount.Looking back at the table, let's list all the ab values:From row 0: 0From row 2: 4,8,12,16,20,24From row 4: 8,16,24,32,40,48From row 6: 12,24,36,48,60,72From row 8: 16,32,48,64,80,96From row 10: 20,40,60,80,100,120From row 12: 24,48,72,96,120,144Now, let's list all these values and count the distinct ones:0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144.That's 19 distinct values. So, ab - 1 would be 19 distinct values.But the initial answer was 28, so I must have made a mistake in my approach.Wait, perhaps I should consider that p and q are positive odd integers, so p and q can be 1, but when p=1, a=0, and when q=1, b=0. So, ab can be 0, but maybe there are other ways to get the same ab - 1 value.Wait, no, because each ab is unique, so ab - 1 should also be unique. So, if ab has 19 distinct values, ab - 1 has 19 distinct values.But the initial answer was 28, so I must have made a mistake in my counting.Wait, let me try a different approach. Maybe I should compute all possible pq - p - q values directly and count the distinct ones.So, let's list all pairs (p, q) where p and q are in {1,3,5,7,9,11,13}, compute pq - p - q for each pair, and then count the distinct results.This will take some time, but let's proceed.First, list all pairs:(1,1), (1,3), (1,5), (1,7), (1,9), (1,11), (1,13),(3,1), (3,3), (3,5), (3,7), (3,9), (3,11), (3,13),(5,1), (5,3), (5,5), (5,7), (5,9), (5,11), (5,13),(7,1), (7,3), (7,5), (7,7), (7,9), (7,11), (7,13),(9,1), (9,3), (9,5), (9,7), (9,9), (9,11), (9,13),(11,1), (11,3), (11,5), (11,7), (11,9), (11,11), (11,13),(13,1), (13,3), (13,5), (13,7), (13,9), (13,11), (13,13).That's 49 pairs. Now, let's compute pq - p - q for each pair.Starting with p=1:(1,1): 1*1 -1 -1 = 1 -1 -1 = -1(1,3): 3 -1 -3 = -1(1,5): 5 -1 -5 = -1(1,7): 7 -1 -7 = -1(1,9): 9 -1 -9 = -1(1,11): 11 -1 -11 = -1(1,13): 13 -1 -13 = -1So, all pairs where p=1 or q=1 result in -1.Now, p=3:(3,1): same as (1,3), which is -1(3,3): 9 -3 -3 = 3(3,5): 15 -3 -5 = 7(3,7): 21 -3 -7 = 11(3,9): 27 -3 -9 = 15(3,11): 33 -3 -11 = 19(3,13): 39 -3 -13 = 23p=5:(5,1): same as (1,5), which is -1(5,3): same as (3,5), which is 7(5,5): 25 -5 -5 = 15(5,7): 35 -5 -7 = 23(5,9): 45 -5 -9 = 31(5,11): 55 -5 -11 = 39(5,13): 65 -5 -13 = 47p=7:(7,1): same as (1,7), which is -1(7,3): same as (3,7), which is 11(7,5): same as (5,7), which is 23(7,7): 49 -7 -7 = 35(7,9): 63 -7 -9 = 47(7,11): 77 -7 -11 = 59(7,13): 91 -7 -13 = 71p=9:(9,1): same as (1,9), which is -1(9,3): same as (3,9), which is 15(9,5): same as (5,9), which is 31(9,7): same as (7,9), which is 47(9,9): 81 -9 -9 = 63(9,11): 99 -9 -11 = 80 - 11? Wait, 99 -9 -11 = 99 -20 = 79Wait, 9*11=99, 99 -9 -11=99-20=79.(9,13): 117 -9 -13 = 117 -22=95p=11:(11,1): same as (1,11), which is -1(11,3): same as (3,11), which is 19(11,5): same as (5,11), which is 39(11,7): same as (7,11), which is 59(11,9): same as (9,11), which is 79(11,11): 121 -11 -11 = 121 -22=99(11,13): 143 -11 -13=143-24=119p=13:(13,1): same as (1,13), which is -1(13,3): same as (3,13), which is 23(13,5): same as (5,13), which is 47(13,7): same as (7,13), which is 71(13,9): same as (9,13), which is 95(13,11): same as (11,13), which is 119(13,13): 169 -13 -13=169-26=143Now, let's list all the results we've got:From p=1 and q=1: -1From p=3:3,7,11,15,19,23From p=5:15,23,31,39,47From p=7:35,47,59,71From p=9:63,79,95From p=11:99,119From p=13:143Now, let's compile all these results:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.Now, let's count the distinct values:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 distinct values.Wait, but earlier I thought the answer was 28. So, where is the discrepancy?Wait, perhaps I made a mistake in the initial transformation. Let me check the transformation again.Original expression: pq - p - q.I rewrote it as (p - 1)(q - 1) - 1.Let me verify with an example:Take p=3, q=3: pq - p - q = 9 -3 -3=3.(p -1)(q -1) -1 = (2)(2) -1=4-1=3. Correct.Another example: p=5, q=7: 35 -5 -7=23.(4)(6) -1=24-1=23. Correct.Another example: p=9, q=11: 99 -9 -11=79.(8)(10) -1=80-1=79. Correct.So, the transformation is correct.Therefore, the number of distinct values should be 19.But the initial answer was 28, so perhaps the initial answer was incorrect.Wait, let me check the problem statement again."Consider two positive odd integers less than 15 (not necessarily distinct). When their product minus the sum of these two numbers is calculated, how many different possible values may result?"So, the problem is correct as stated.Wait, but in my calculation, I got 19 distinct values, but the initial answer was 28. So, perhaps I made a mistake in my counting.Wait, let me recount the distinct values from the list:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 values.Wait, but when I listed the results from each p, I had:From p=1: -1From p=3: 3,7,11,15,19,23From p=5:15,23,31,39,47From p=7:35,47,59,71From p=9:63,79,95From p=11:99,119From p=13:143So, let's list them all:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.Yes, that's 19 distinct values.Therefore, the correct answer should be 19.But the initial answer was 28, which suggests a mistake in the initial reasoning.Wait, perhaps the initial answer considered that when p=1, q=1, the result is -1, but when p=1 and q=3, it's also -1, and so on. So, all pairs where either p or q is 1 result in -1. So, there are 13 such pairs (7 with p=1 and 6 with q=1, but (1,1) is counted once). So, 13 pairs result in -1.Then, the remaining 49 -13=36 pairs result in other values.But in my calculation, the distinct results are 19, including -1.Wait, perhaps the initial answer considered that when p and q are both greater than 1, the number of distinct results is 28, but that doesn't make sense because the total number of pairs is 49, and many of them result in the same value.Wait, perhaps the initial answer was incorrect, and the correct number is 19.Alternatively, maybe I made a mistake in my counting.Wait, let me list all the results again:From p=1: -1From p=3: 3,7,11,15,19,23From p=5:15,23,31,39,47From p=7:35,47,59,71From p=9:63,79,95From p=11:99,119From p=13:143Now, let's list all the results:-1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 distinct values.Therefore, the correct answer should be 19.But the initial answer was 28, so perhaps the initial answer was wrong.Alternatively, maybe I made a mistake in my approach.Wait, perhaps I should consider that when p and q are both greater than 1, the number of distinct results is higher.Wait, but in my calculation, even when considering all pairs, including those with p=1 or q=1, the total distinct results are 19.Therefore, I think the correct answer is 19.But the initial answer was 28, so perhaps I need to double-check.Wait, perhaps I made a mistake in the transformation.Wait, let me consider that pq - p - q = (p -1)(q -1) -1.So, if I let a = p -1 and b = q -1, then the expression is ab -1.Now, a and b can be 0,2,4,6,8,10,12.So, ab can be 0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144.So, ab -1 can be -1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 distinct values.Therefore, the correct answer is 19.But the initial answer was 28, so perhaps the initial answer was incorrect.Alternatively, perhaps the problem was misstated.Wait, the problem says "two positive odd integers less than 15", so p and q are in {1,3,5,7,9,11,13}.So, a and b are in {0,2,4,6,8,10,12}.Therefore, ab can be 0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144.So, ab -1 can be -1,3,7,11,15,19,23,31,35,39,47,59,63,71,79,95,99,119,143.That's 19 distinct values.Therefore, the correct answer is 19.But the initial answer was 28, so perhaps the initial answer was wrong.Alternatively, perhaps I made a mistake in my counting.Wait, let me check the number of distinct ab values again.From the table:0,4,8,12,16,20,24,32,36,40,48,60,64,72,80,96,100,120,144.That's 19 distinct ab values.Therefore, ab -1 is 19 distinct values.Therefore, the correct answer is 19.But the initial answer was 28, so perhaps the initial answer was incorrect.Alternatively, perhaps the problem was to find the number of distinct values of pq - p - q without considering the transformation, leading to a different count.But in my detailed calculation, I found 19 distinct values.Therefore, I think the correct answer is 19.

💡First, determine the work rates of A and B. A can complete the work in 12 days, so A's work rate is ( frac{1}{12} ) per day. B can complete the work in 6 days, so B's work rate is ( frac{1}{6} ) per day.Next, calculate the amount of work A completes in the first 3 days: ( 3 times frac{1}{12} = frac{1}{4} ).Subtract this from the total work to find the remaining work: ( 1 - frac{1}{4} = frac{3}{4} ).Add A's and B's work rates to find their combined work rate: ( frac{1}{12} + frac{1}{6} = frac{1}{4} ).Finally, divide the remaining work by the combined work rate to find the number of days needed: ( frac{3}{4} div frac{1}{4} = 3 ) days.

💡First, determine the total time Ethan spent learning the 12 fencing tricks. Alexa's vacation duration is 1 week and 2 days, which equals 9 days. Since Alexa's vacation was 3/4 of Ethan's learning time, we can set up the equation:9 days = (3/4) * Ethan's timeSolving for Ethan's time:Ethan's time = 9 days / (3/4) = 12 daysNext, Joey spent 6 days learning to swim. To find the fraction of Ethan's time that Joey spent, divide Joey's time by Ethan's time:Fraction = 6 days / 12 days = 1/2Therefore, Joey spent half of the time Ethan took to learn the fencing tricks.