- #1

Math100

- 756

- 205

- Homework Statement
- The three most recent appearances of Halley's comet were in the years ## 1835, 1910 ##, and ## 1986 ##; the next occurrence will be in ## 2061 ##. Prove that ## 1835^{1910}+1986^{2061}\equiv 0\pmod {7} ##.

- Relevant Equations
- None.

Proof:

Observe that ## 1835\equiv 1\pmod {7}\implies 1835^{1910}\equiv 1\pmod {7} ##.

Then ## 1986\equiv 5\pmod {7} ##.

Applying the Fermat's theorem produces:

## 5^{6}\equiv 1\pmod {7} ##.

This means ## 1986^{2061}\equiv 5^{6\cdot 343+3}\pmod {7}\equiv 5^{3}\pmod {7}\equiv 6\pmod {7} ##.

Thus ## 1835^{1910}+1986^{2061}\pmod {7}\equiv (1+6)\pmod {7}\equiv 0\pmod {7} ##.

Therefore, ## 1835^{1910}+1986^{2061}\equiv 0\pmod {7} ##.

Observe that ## 1835\equiv 1\pmod {7}\implies 1835^{1910}\equiv 1\pmod {7} ##.

Then ## 1986\equiv 5\pmod {7} ##.

Applying the Fermat's theorem produces:

## 5^{6}\equiv 1\pmod {7} ##.

This means ## 1986^{2061}\equiv 5^{6\cdot 343+3}\pmod {7}\equiv 5^{3}\pmod {7}\equiv 6\pmod {7} ##.

Thus ## 1835^{1910}+1986^{2061}\pmod {7}\equiv (1+6)\pmod {7}\equiv 0\pmod {7} ##.

Therefore, ## 1835^{1910}+1986^{2061}\equiv 0\pmod {7} ##.