Same Remainder Iff Divisor Divides Difference

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Problem 45: (Same Remainder Iff Divisor Divides Difference)

Let $a_1,a_2,b\in \mathbb{Z}$ and suppose $b>0$. We can use the division algorithm to write $a_1=q_1b+r_1$ and $a_2=q_2b+r_2$ for some integers $q_1,q_2,r_1,r_2$ with $0\leq r_1<b$ and $0\leq r_2<b$ . Prove that $r_1=r_2$ if and only if $b$ divides $a_1-a_2$.

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