Question

Linear congruence.
\[
3x+y\equiv21 \pmod 5\text{ and }x-y\equiv7\pmod 5
\]
Find the smallest positive value of a such that \(x\equiv a\pmod 5\).

Answer 1

Can I rearrange the equation and subtract two equations together?

Answer 2

Just want to make sure that I can subtract congruence relationships together.

Answer 3

@ganeshie8

Answer 4

definitely yes, you can subtract/add :
If \(a\equiv b \pmod{n}\) and \(c\equiv d\pmod{n}\), then
\(a+c \equiv b+d \pmod n\)

Answer 5

Another question.

Answer 6

Given that p is a prime number show that \(p|(ab^p-a^pb)\) for all positive integers a and b.

Answer 7

familiar with Fermat's little thm ?

Answer 8

Nope. All I know is \(a^p\equiv a\pmod p\).

Answer 9

thats Fermat's little thm and yes thats all we need to know

Answer 10

\(a^p \equiv a \pmod p\)
\(b^p \equiv b \pmod p\)

Answer 11

multiply first congruence by \(b\)
multiply second congruence by \(a\)

subtract