I have recently come across this problem and am having trouble proving they are equal sets. I can prove that the second set is a subset of the first through plugging in (8x-14) into x, but can't find a good way to prove the first set is a subset of the second to prove their equality.
$$\{x\in \mathbb{Z} : 7|x\} = \{x\in \mathbb{Z} : 7|(8x-14)\}$$
Any help would be appreciated!
Best Answer
Hint. $$8x-14=x+7x-14=x+7(x-2)$$