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Number Theory

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Every Integer Is of the Form...

NUMTH-D51GRS

The division algorithm states that given any $a \in \mathbb{Z}$ and $d \in \mathbb{N}$ there are unique $q, r \in \mathbb{Z}$ such that $a = dq + r$ with $0 \leq r < d$. Which of the following statements are true thanks to the division algorithm?

Select ALL that apply.

A

Every integer is of the form $2j$ or $2j + 1$ for some $j \in \mathbb{Z}$.

B

Every integer is of the form $4m$ or $4m + 1$ for some $m \in \mathbb{Z}$.

C

Every integer is of the form $3k, 3k + 1$ or $3k + 2$ for some $k \in \mathbb{Z}$.

D

Every integer is of the form $5n + 1, 5n + 2, 5n + 3$ or $5n + 4$ for some $n \in \mathbb{Z}$.