workout wrote:
If k is a positive integer, what is the smallest possible value of k such that 1040k is the square of an integer?
A) 2
B) 5
C) 10
D) 15
E) 65
Questions like this can often be solved by figuring out prime factors.
We know that 1040 * k is a perfect square, so find the prime factorization of 1040 first:
104*10
2*52* 5*2
2*2*26*5*2
2*2*2*13*5*2
So the prime factorization of 1040 = \(2^4*5*13\)
For a number to be a perfect square, each of its prime factors need to be paired with a matching prime factor.
\(2^4\) is 16, a perfect square. Each factor of 2 is paired with another factor of 2. But 5 and 13 don't have matching factors, so we need another 5 * 13 to make a perfect square. That product is k.
k = 5*13 = 65
Answer: E