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Number Theory

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Moderate

Multiplicative Order: Possibility

NUMTH-BK9PM3

Denote by $\mathrm{ord}_p (a)$ the multiplicative order of $a$ modulo $p$, if it exists. This means that:

$$ \mathrm{ord}_p(a) = d \ \Longleftrightarrow \ a^d \equiv 1 \ \mathrm{mod} \ p, \mathrm{and} \ a^{d'}\not\equiv 1 \ \mathrm{mod} \ p, \ \mathrm{for} \ 0 < d' < d $$

Let $p=101$. Which of the following integers can be $\mathrm{ord}_p(a)$ for some integer $a$?

A

1

B

3

C

25

D

100