- #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} ##.