Problem 30 · 2013 Math Kangaroo
Stretch
Number Theory
divisibilityfactorization
A positive integer N is smaller than the sum of its three biggest proper factors (N itself is not a proper factor of N). Which of the following statements is true?
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Answer: B — All such numbers N are divisible by 6.
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Hint 1 of 2
The biggest proper factors of N are N/2, N/3, N/4, ... — write the sum of the top three this way.
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Hint 2 of 2
For that sum to exceed N, the small divisors 2 and 3 must both divide N.
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Approach: express the largest proper factors as N over its smallest divisors
- The three biggest proper factors are N divided by its three smallest divisors (other than 1).
- Their sum can exceed N only when both 2 and 3 divide N (for example N/2 + N/3 + N/4 = 13N/12 > N).
- So every such N is divisible by 6.
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