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Thursday, March 3, 2016

Problems in Number Theory

Problem:


Solution:

The key step in this problem is to let $ n = A2^p3^q5^r $, it can be easily seen that $ A = 1 $ for minimal value in the set.

The fact that these roots are natural number give rise to these equations:

$ p - 1 = 0 \mod 2 $
$ p = 0 \mod 3 $
$ p = 0 \mod 5 $

$ q = 0 \mod 2 $
$ q - 1 = 0 \mod 3 $
$ q = 0 \mod 5 $

$ r = 0 \mod 2 $
$ r = 0 \mod 3 $
$ r - 1 = 0 \mod 5 $

By Chinese remainder theorem, we get $ p = 15 $, $ q = 10 $, $ r = 6 $, so the answer is

$ 2^15 \times 3^10 \times 5^6 = 30,233,088,000,000 $

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