Question

If \(\gcd(a,30)=1\), show that \(60\) divides \(a^4+59\)

Answer 1

60 = 2^2 . 3 . 5

Answer 2

\(\large a\) is not even, so

\(\large a \equiv 1 \pmod 2 \implies a^2 \equiv 1 \pmod 4 \implies a^4 \equiv 1 \pmod 4\)

Answer 3

Hint :-
a is odd thus

a=1 mod 4
or
a=-1 mod 4
both cases gives

a^4=1 mod 4

now
a^4=1 mod 4 (Fermat )
a^2=1 mod 3
a^4=1 mod 3

Answer 4

got it, thanks!

Answer 5

not welcome

Answer 6

i made a typo with next step
a^4=1 mod 5

Answer 7

How is below argument without using odd/even thingy :

\(\large a = 1 \pmod 2\\ \implies a^2 = 1 \pmod 2 \\ \implies a^2-1 \equiv 0 \pmod 2 \\ \implies (a+1)(a-1)\equiv 0 \pmod 2\)

Answer 8

a is odd , so a-1 and a+1 both are even

Answer 9

Exactly! so both are divisible by 2 and hence (a+1)(a-1) = 0 mod 2^2

Answer 10

yep.