determine whether each of the following "theorems" is true or false. Assume that a, b, c, d,
and m are integers with m 1.
-$$\text { If } a \equiv b ( \bmod m ) \text {, and } a \equiv c ( \bmod m ) \text {, then } a \equiv b + c ( \bmod m ) \text {. }$$

Question

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d,
and m are integers with m > 1.
-$$\text { If } a \equiv b ( \bmod m ) \text {, and } a \equiv c ( \bmod m ) \text {, then } a \equiv b + c ( \bmod m ) \text {. }$$

Quizplus · Accepted Answer

The correct conclusion from the given premises is that $b \equiv c \mod m$, not that $a \equiv b + c \mod m$. The given premises only ensure that $a$ and $b$ leave the same remainder when divided by $m$, and $a$ and $c$ leave the same remainder when divided by $m$, but this does not imply that $a$ is congruent to $b+c$ modulo $m$.

Determine Whether Each of the Following "Theorems" Is True or False