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07 Nov 2004, 11:43
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How many integers from 1 to 2003 inclusive have an odd number of distinct positive divisors?
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21 Jul 2011, 06:24
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Tricky. Before jumping in, you should consider a couple of examples:
24 = 1 * 24
24 = 2 * 12
24 = 3 * 8
24 = 4 * 6
Try a few other examples and the pattern should become clear - divisors come in pairs, so the number of divisors will always be even - UNLESS one of the divisors is repeated. The only way one of the divisors will be repeated is if the original number is a square. E.g. the divisors of 4 are 1, 2, and 4 - you don't count 2 twice because the question asks for distinct divisors.
So, the question is really this, how many squares are less than or equal to 2003 and greater than or equal to 1.
Mental math - the root of 2003 will be between 40 and 50 probably closer to 40.
Try 42: 42^2 1764. Try 44: 44^2 = 1936. Check 45: 45^2 = 2025 - too big.
So, the last square less than or equal to 2003 is 44^2. Thus there are 44 numbers between 1 and 2003 inclusive with an odd number of distinct divisors.
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