- #1

Math100

- 782

- 220

- Homework Statement
- Establish the following divisibility criteria:

An integer is divisible by ## 4 ## if and only if the number formed by its tens and units digits is divisible by ## 4 ##.

[Hint: ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##.]

- Relevant Equations
- None.

Proof:

Let ## N ## be an integer.

Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##.

Note that ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##.

Thus ## 4\mid N\Leftrightarrow N\equiv 0\pmod {4}\Leftrightarrow a_{1}10+a_{0}\equiv 0\pmod {4} ##.

Therefore, an integer is divisible by ## 4 ## if and only if the number formed by its tens and units digits is divisible by ## 4 ##.

Let ## N ## be an integer.

Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##.

Note that ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##.

Thus ## 4\mid N\Leftrightarrow N\equiv 0\pmod {4}\Leftrightarrow a_{1}10+a_{0}\equiv 0\pmod {4} ##.

Therefore, an integer is divisible by ## 4 ## if and only if the number formed by its tens and units digits is divisible by ## 4 ##.