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Number Theory

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Moderate

What Is True If $a$ Does Not Divide $b$?

NUMTH-SXQR5A

Let $a \in \mathbb{Z}$ and $b \in \mathbb{N}$. What is true if $a \nmid b$ ($a$ does not divide $b$)?

A

There exist unique $q, r \in \mathbb{N}$ such that $b^2 = a^2 q^2 + r^2$ and $0 < r < a$.

B

There exist unique $q, r \in \mathbb{Z} \backslash \{0 \}$ such that $b = (aq)^{r}$ and $0 < r < a$.

C

There exist unique $q, r \in \mathbb{N}$ such that $b = aq - r$ and $0 < r < a$.

D

There exist unique $q, r \in \mathbb{Z}$ such that $e^{b - r} = e^{aq}$ and $0 < r < a$.