This problem is so last year!

Number Theory Level 4

If a b × b c a^{b}\times b^{c} is a multiple of 2016, where a a , b b and c c are positive integers, find the least possible value of a + b + c a+b+c .


The answer is 14.

Freddie Hand by Freddie Hand , 18, United Kingdom

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1 solution

We note that 2016 = 2 5 × 3 2 × 7 2016=2^5 \times 3^2 \times 7 in terms of prime factors. Arranging a b b c = 6 7 × 7 1 = 2 2 × 3 5 × 2016 a^bb^c = 6^7\times 7^1 = 2^2 \times 3^5 \times 2016 , we get the least possible a + b + c = 6 + 7 + 1 = 14 a+b+c = 6+7+1 = \boxed{14} .

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