Exercise 223Number theory › CongruencesHigh schoolAdded 3 Nov 2026

Arithmetic modulo $n$

#modular arithmetic#operations#compatibility

Statement

Compute the following in $\mathbb{Z}/7\mathbb{Z}$ :
(a) $3+5$ ; (b) $4\cdot6$ ; (c) $2^5$ ; (d) $3-6$ .

Hint

Reduce modulo $7$ at each step.

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(a) $3+5=8\equiv1\pmod7$ .

(b) $4\cdot6=24\equiv3\pmod7$ .

(c) $2^5=32\equiv4\pmod7$ (since $28=4\cdot7$ ).

(d) $3-6=-3\equiv4\pmod7$ .

Conclusion. The results in $\mathbb{Z}/7\mathbb{Z}$ are $\bar1,\bar3,\bar4,\bar4$ .