- #1

Math100

- 792

- 220

- Homework Statement
- Assuming that ## a ## and ## b ## are integers not divisible by the prime ## p ##, establish the following:

If ## a^{p}\equiv b^{p}\pmod {p} ##, then ## a\equiv b\pmod {p} ##.

- Relevant Equations
- None.

Proof:

Suppose ## a^{p}\equiv b^{p}\pmod {p} ##, where ## a ## and ## b ## are integers not divisible by the prime ## p ##.

Then ## p\nmid a ## and ## p\nmid b ##.

Applying the Fermat's theorem produces:

## a^{p}\equiv a\pmod {p}, b^{p}\equiv b\pmod {p} ##.

Thus ## a^{p}\equiv a\equiv b^{p}\equiv b\pmod {p} ##.

Therefore, if ## a^{p}\equiv b^{p}\pmod {p} ##, then ## a\equiv b\pmod {p} ##.

Suppose ## a^{p}\equiv b^{p}\pmod {p} ##, where ## a ## and ## b ## are integers not divisible by the prime ## p ##.

Then ## p\nmid a ## and ## p\nmid b ##.

Applying the Fermat's theorem produces:

## a^{p}\equiv a\pmod {p}, b^{p}\equiv b\pmod {p} ##.

Thus ## a^{p}\equiv a\equiv b^{p}\equiv b\pmod {p} ##.

Therefore, if ## a^{p}\equiv b^{p}\pmod {p} ##, then ## a\equiv b\pmod {p} ##.