To find the smallest possible value of k such that 1040k is the square of an integer, we need to factorize 1040 and then find the least value of k that makes the result a perfect square.

First, let's factorize 1040:

1040 = 2^4 * 5 * 13

Now, to make 1040k a perfect square, we need to make the exponents of all prime factors even. So, we need to find the least value of k that will make the exponents of 2, 5, and 13 even.

Exponent of 2: Since it's already raised to the power of 4, we don't need to change it.

Exponent of 5: We need to raise it to an even power. So, k should be multiplied by 5 to make it even.

Exponent of 13: We need to raise it to an even power. So, k should be multiplied by 13 to make it even.

Now, let's consider the exponents of 5 and 13:

Exponent of 5: It's currently 1, so we need to raise it to an even power, which means multiplying by 5 again. So, k should be at least 5.

Exponent of 13: It's currently 1, so we need to raise it to an even power, which means multiplying by 13 again. So, k should be at least 13.

To make sure both exponents are even, we need to take the least common multiple (LCM) of 5 and 13, which is 65.

So, the smallest possible value of k such that 1040k is the square of an integer is k = 65.