A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same as "9 and 16".)
you want x^2-y^2 = 495
(x+y)(x-y) = 495
495 = 3^2*5*11
so, the factors are
1 3 5 9 11 15 33 45 55 99 165 495
so, if you have
x+y=495
x-y=1
248^2-247^2 = 495
x+y=165
x-y=3
84^2-81^2 = 495
and so on
please continue
nope. Just take the factors in pairs, as I did, working your way in from the ends.
Actually, even that's not necessary, unless you just want to verify the results. There are 12 factors, so that means there are 6 pairs like the two I showed above.
So, there are 6 pairs of numbers whose squares differ by 495.
To find the pairs of perfect squares that differ by 495, we need to consider two perfect squares whose difference is 495.
Let's assume the two perfect squares are x^2 and y^2, where x and y are positive integers.
According to the problem, the difference between these two perfect squares is 495, so we can set up the equation:
x^2 - y^2 = 495
Now, notice that the left side of the equation is a difference of squares. We can factor it using the difference of squares formula:
(x - y)(x + y) = 495
We need to find factors of 495 such that the resulting expressions (x - y) and (x + y) are both perfect squares.
To do this, let's find the prime factorization of 495:
495 = 3^2 * 5 * 11
Now, we can try to pair these prime factors in different ways. Since (x - y) and (x + y) must be perfect squares, they should have the same prime factors, with each factor appearing an even number of times.
Let's examine the possibilities:
1. (3^2) * (5^2) = (3 * 5)^2
Here, (x - y) = 3 * 5 = 15 and (x + y) = (x - y) + 2y = 15 + 2y
We can solve this system of equations to find the value of y.
2. (3^2) * (5^2 * 11) = 3^2 * (5 * 11)^2
Here, (x - y) = 3 and (x + y) = 5 * 11 = 55
Again, we can solve this system of equations to find the value of y.
3. (3^2 * 5) * 11^2 = (3 * 11)^2 * 5
Here, (x - y) = 3 * 11 = 33 and (x + y) = (x - y) + 2y = 33 + 2y
We can solve this system of equations to find the value of y.
By solving these systems of equations, we can determine the values of x and y for each case. If x and y are both positive integers, then we have found a pair of perfect squares that differ by 495.
Now, it's your turn to solve these systems of equations and find the values of x and y for each case. Once you have found the values of x and y, count the number of pairs of perfect squares you obtained.