- #1

Math100

- 792

- 220

- Homework Statement
- Establish the following divisibility criteria:

An integer is divisible by ## 2 ## if and only if its units digit is ## 0, 2, 4, 6 ##, or ## 8 ##.

- Relevant Equations
- None.

Proof:

Suppose ## N ## is the integer and ## x ## is the units digit of ## N ##.

Then ## N=10k+x ## for some ## k\in\mathbb{Z} ## where ## x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ##.

Note that ## 10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2} ##.

Thus ## 2\mid N\implies N\equiv 0\pmod {2}\implies x\equiv 0\pmod {2}\implies x\in{0, 2, 4, 6, 8} ##.

Therefore, an integer is divisible by ## 2 ## if and only if its units digit is ## 0, 2, 4, 6 ##, or ## 8 ##.

Suppose ## N ## is the integer and ## x ## is the units digit of ## N ##.

Then ## N=10k+x ## for some ## k\in\mathbb{Z} ## where ## x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ##.

Note that ## 10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2} ##.

Thus ## 2\mid N\implies N\equiv 0\pmod {2}\implies x\equiv 0\pmod {2}\implies x\in{0, 2, 4, 6, 8} ##.

Therefore, an integer is divisible by ## 2 ## if and only if its units digit is ## 0, 2, 4, 6 ##, or ## 8 ##.