- #1

Math100

- 759

- 206

- Homework Statement
- If ## 7\nmid a ##, prove that either ## a^{3}+1 ## or ## a^{3}-1 ## is divisible by ## 7 ##.

- Relevant Equations
- None.

Proof:

Suppose ## 7\nmid a ##.

Applying the Fermat's theorem produces:

## a^{7-1}\equiv 1\pmod {7}\implies a^{6}\equiv 1\pmod {7} ##.

This means ## 7\mid (a^{6}-1) ##.

Observe that ## a^{6}-1=(a^{3}-1)(a^{3}+1) ##.

Thus ## 7\nmid (a^{3}-1)\implies 7\mid (a^{3}+1) ## and ## 7\nmid (a^{3}+1)\implies 7\mid (a^{3}-1) ##.

Therefore, if ## 7\nmid a ##, then either ## a^{3}+1 ## or ## a^{3}-1 ## is divisible by ## 7 ##.

Suppose ## 7\nmid a ##.

Applying the Fermat's theorem produces:

## a^{7-1}\equiv 1\pmod {7}\implies a^{6}\equiv 1\pmod {7} ##.

This means ## 7\mid (a^{6}-1) ##.

Observe that ## a^{6}-1=(a^{3}-1)(a^{3}+1) ##.

Thus ## 7\nmid (a^{3}-1)\implies 7\mid (a^{3}+1) ## and ## 7\nmid (a^{3}+1)\implies 7\mid (a^{3}-1) ##.

Therefore, if ## 7\nmid a ##, then either ## a^{3}+1 ## or ## a^{3}-1 ## is divisible by ## 7 ##.