I'm not sure how to do this:
show that $(x + 12y)^{13}\equiv x^{13} - y^{13} {\mod 13}$ for all integers
It's part of a question on the binomial theorem, but that's all I've got.
I'm not sure how to do this:
show that $(x + 12y)^{13}\equiv x^{13} - y^{13} {\mod 13}$ for all integers
It's part of a question on the binomial theorem, but that's all I've got.
Hint 1: $(x + 12y)^{13}\equiv (x - y)^{13} {\mod 13}$
Hint 2: in the expansion of $(a + b)^{p}$ for $p$ prime, each term has a co-efficient which is a multiple of $p$ except the first and last.
Hint :
If we apply freshman dream theorem we can write :
$(x+12y)^{13} \equiv x^{13}+(12y)^{13} \pmod {13}$
Now , use a fact :
$12 \equiv -1 \pmod {13} \Rightarrow 12^{13} \equiv -1 \pmod{13}$