Notice that
The group has property than anything raised to gives the identity element. Therefore, if then . It follows that i.e.
Hi all,
I am kind of stuck with these proofs, which seemingly involve FLT ultimately. Since they're more or less related, I'm posting all three:
Let a be some integer. Prove that (let == be the symbol for congruence):
- a^21 == a (mod 15)
- a^7 == a (mod 42)
- if gcd(a,35) = 1 (i.e., a and 35 are coprimes), then a^12 == 1 (mod 35)
After tinkering with the problems a bit, I suspect that the solution involves the GCD of the exponent and the modulus. So, more specifically, I'd like help specially in rewriting the congruences so that the modulus is prime, while keeping it true. That the modulus is prime is obviously required by the FLT hypotesis.
Any help or tip for one or more of the proofs will be greatly appreciated. Thanks in advance.
first note that if then
by FLT: but: hence:
by FLT: but: hence:
now (1) and (2) complete the proof.
by FLT: again by FLT: and so:a^7 == a (mod 42)
since by FLT we have: but thus:if gcd(a,35) = 1 (i.e., a and 35 are coprimes), then a^12 == 1 (mod 35)
since by FLT we have: but: thus:
now (1) and (2) complete the proof.
Q.E.D.