Modular arithmetic

Number Theory Level 3

Let $a$ be a natural number. Which of the following is equivalent to $a^7$ when considered modulo 42?

a^2

a

None

2a

1 solution

Mathh Mathh
Apr 15, 2015

$a^7\equiv a^{7\pmod {6}}\equiv a\pmod {7}$ by Fermat's Little Theorem. Similarly:

$a^7\equiv a^{7\pmod {2}}\equiv a\pmod {3}$

$a^7\equiv a^{7\pmod {1}}\equiv a\pmod {2}$

These three statements imply $2,3,7\mid a^7-a\iff 42\mid a^7-a\iff a^7\equiv a\pmod {42}$ .