Monday, 6 June 2016

Arithmetic of integers: divisibility, modular congruence.

Show that 70 divide 1016n1 for all n natural numbers.



I tried to show that 1016n1 mod 70.



Thanks for all. I got it.



Note that 70 = 7.5.2




As 101ϕ(7) 1 mod 7, so 1016n 1 mod 7 (Euler Theorem)



and 101ϕ(2) 1 mod 2, so 1016n 1 mod 2 (Euler Theorem)



and 101 1 mod 5, so 1016n 1 mod 5



Therefore, 1016n 1 mod 70.

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