Question

how many integers between 1 to 1000 can't be expressed as a difference between the squares of two integers​

Answer 1

The number of integers that can't be expressed as a difference between the squares of two integers is 250.

• This can be calculated by first calculating the number of integers that can be expressed as a difference between the squares of two integers. This is done as follows:

• Let us consider Case 1 in which both the numbers are even (2a, 2b)

        ∴ 4a² - 4b² = 4(a² -b²) = (4k)

• Let us consider Case 2 where both numbers are odd (2a + 1 , 2b + 1)

       ∴ 4a² + 4a+ 1 - 4b² - 4b - 1 = 4(a² + a - b² - b) = (4k)

• Let us consider Case 3 where first one is even and second one is odd (2a, 2b + 1)

        ∴ 4a² - 4b² - 4b - 1 = 4k - 1 or

                                        = (4k + 3)

• Let us consider Case 4 in which first one is odd and second one is even (2a + 1, 2b)

        ∴ 4a² + 4a+ 1 - 4b² = (4k + 1)

• This shows that the difference of squares of two non negative integers can be (4k), (4k+1) or (4k + 3) but can never be (4k + 2).

• So, out of 4 cases, 3 will be used

       ⇒ (3/4)(1000) = 750

• Thus 750 can be expressed as a difference between the squares of two integers.

• For numbers that can't be expressed as a difference between the squares of two integers,

        1000 - 750 = 250 numbers