determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m 1.
-$$\text { If } a \equiv b ( \bmod m ) \text {, then } a \equiv b ( \bmod 2 m ) \text {. }$$

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.
-$$\text { If } a \equiv b ( \bmod m ) \text {, then } a \equiv b ( \bmod 2 m ) \text {. }$$

Quizplus · Accepted Answer

If $ a \equiv b \pmod{m} $, it means $ m \mid (a-b) $. However, this does not imply $ 2m \mid (a-b) $, so $ a \equiv b \pmod{2m} $ is not guaranteed.

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