Now, for a number to be the square of an integer, there must be an equal number of all prime factors. So we can split up these factors so that there's an equal number of each (or as close to it as possible):
1040k = (2 x 2 x 5 x 13)(2 x 2 x k)
This tells us that k must equal 5 x 13, since that will make both terms equal, resulting in a perfect square. k = 5 x 13 = 65: choice (E).
FAQ: Where do those numbers that make up 1040 come from? How can you do that quickly?
This is what's known as prime factorization. You can do it more quickly by pulling out one factor at a time. It's easiest to first pull out a ten (2 * 5), and then pull out factors of 2 until you can no longer do so. Here are the full steps:
1040 = 10 * 104
1040 = 2 * 5 * 104
1040 = 2 * 5 * 2 * 52
1040 = 2 * 5 * 2 * 2 * 26
1040 = 2 * 5 * 2 * 2 * 2 * 13
This rearranges to
1040 = 2^4 * 5 * 13
FAQ: I interpreted k as being the last digit of the number, not as 1040 * k. How am I supposed to know which it is?
Since it's not stated that k is the last digit of 1040k, we can assume that 1040 is multiplied by k, just like how 10x is 10 * x, or 5y is 5 * y. If k would be the last digit of 1040k, the question would need to explicitly say that. If not, you can just assume it's 1040 * k.
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