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By saba_tavdgiridze, 7 years ago, In English

A positive integer N is smaller than the sum of its three greatest divisors (naturally, excluding N itself). Which of the following statements is true? (A) All such N are divisible by 4. (B) All such N are divisible by 5.
(C) All such N are divisible by 6. (D) All such N are divisible by 7. (E) There is no such N.

 
 
 
 
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7 years ago, # |
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The three biggest divisors are N divided by the three smallest divisors. Let's suppose the three smallest divisors are d1, d2 and d3 (d1 < d2 < d3), then the following inequality holds...

N / d1 + N / d2 + N / d3 > N

N * (1 / d1 + 1 / d2 + 1 / d3) > N

1 / d1 + 1 / d2 + 1 / d3 > 1

Let's suppose that d1 > 2, then the maximum value that the summand can take is 1 / 3 + 1 / 4 + 1 / 5 = 47 / 60, which is less than 1. So now we know that d1 is 2. Let's see what happens when d2 > 3. The maximum value the summand can take is 1 / 2 + 1 / 4 + 1 / 5 = 19 / 20. So we know as well that d2 is 3. Knowing the values of d1 and d2, let's find out the possible values of d3...

1 / 2 + 1 / 3 + 1 / d3 > 1

1 / d3 > 1 - 1 / 2 - 1 / 3

1 / d3 > 1 / 6

d3 < 6

Finally, d3 can be either 4 or 5. So N is of the form 22 * 3 * k or 2 * 3 * 5 * k. In either case, N is a multiple of 6. So the correct answer is C.