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Which of the following is/are True?

1. Suppose $na\equiv nb\; \mod\; m,$ then $a\equiv b \;\mod\; m$ holds
2. $x^3$ is always congruent to one of $-1, 0, 1$ on $\mod\; 7$.
3. Suppose $a\equiv b\; \mod \;m$ and $a'\equiv b' \; \mod \;m,$ then $aa'\equiv bb'\; \mod\; m$
4. Suppose $a\equiv b \; \mod\; m,$ then $a+m \equiv b \;\mod\; m$

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@GO Classes edit necessary

## 1 Answer

$\text{A}$ is false because we can not cancel any arbitrary number.

$\text{B}$ is true as shown below.

$0^3\;\equiv \; 0 \ (\mod \; 7)$
$1^3\;\equiv \; 1 \; (\mod \; 7)$
$2^3\;\equiv \; 1 \; (\mod \; 7)$
$3^3\;\equiv \; -1 \; (\mod \; 7)$
$4^3\;\equiv \; 1 \; (\mod \; 7)$
$5^3\;\equiv \; -1 \; (\mod \; 7)$
$6^3\;\equiv \; -1 \; (\mod \; 7)$

If we take values greater than $7$, it will lead to one of the above values.

For example $- 83\equiv 13 \equiv 1 \; \mod \; 7 \text{ or } 93 \equiv 23 \equiv 1 \; \mod \;7$
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