Problem 25 · 2009 Math Kangaroo
Stretch
Number Theory
factorizationprimes
All factors of a number N (with the exception of 1 and N itself) are written down one after the other. It turns out that the biggest factor is 45 times as big as the smallest factor. For how many numbers N is that true?
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Answer: C — 2
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Hint 1 of 2
The smallest factor above 1 is the least prime p; the largest below N is N/p.
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Hint 2 of 2
Set N/p = 45p, so N = 45 p^2, and check p is really the smallest prime of N.
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Approach: translate the factor condition into N = 45 p^2
- The condition (largest proper factor) = 45 x (smallest proper factor) gives N/p = 45p, so N = 45 p^2.
- Because 45 = 3^2 x 5, N always has factor 3, so the smallest prime p can only be 2 or 3: p=2 gives N=180, p=3 gives N=405, both valid.
- Hence there are 2 such numbers.
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