The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y?8085115165下一题 Next

To solve this problem, we need to find the smallest x and y such that 360x is a perfect square and 360y is a perfect cube.

First, let's factorize 360. The prime factorization of 360 is 2^3 * 3^2 * 5.

For a number to be a perfect square, all the exponents in its prime factorization must be even. For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

For 360x to be a perfect square, we need to multiply 360 by a number that will make all the exponents even. The smallest such number is 2 * 5 = 10. So, x = 10.

For 360y to be a perfect cube, we need to multiply 360 by a number that will make all the exponents multiples of 3. The smallest such number is 2 * 3 * 5^2 = 30. So, y = 30.

Therefore, the sum of x and y is 10 + 30 = 40.