Maths Olympiad Prep

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Number theory Difficulty 6.2 National olympiad Prove it

20. Prove: For any integer nn, at least one of the following six congruences holds:
n≡0( mod 2),n≡0( mod 3),n≡1( mod 4)n≡3( mod 8),n≡7( mod 12),n≡23( mod 24)\begin{array}{lll} n \equiv 0(\bmod 2), & n \equiv 0(\bmod 3), & n \equiv 1(\bmod 4) \\ n \equiv 3(\bmod 8), & n \equiv 7(\bmod 12), & n \equiv 23(\bmod 24) \end{array}

Solution

20. As in the proof of the previous problem.

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