When $A$ and $B$ are divided by $173$, the remainder is $33$ and $17$ respectively. Find the remainder when $2A+B$ is divided by $173.$

Question

Question: When positive integer $A$ is divided by $173$, the remainder is $33$, and when positive integer $B$ is divided by $173$, the remainder is $17$. What is the remainder when $2A+B$ is divided by $173$?

This is GRE quant question, and the solution says that it is valid to assume that both $A$ and $B$ have the same quotient. Then, $A = B + 16$.

$2(A)+B = 2(B+16)+B = 3B+32$. Therefore, the remainder is $17\cdot 3 + 32 = 83$.

But, I don't understand the assumption above. How can we assume that $A$ and $B$ have the same quotient??

Thank you for helping in advance.

Answer 1

Strictly speaking

$$A= 173Q_1 + 33$$ $$B = 173Q_2 + 17$$

then

$$2A+B= 2(173Q_1+33) + 173Q_2+17=173(2Q_1+Q_2)+2(33)+17$$

The answer is $2(33)+17$.

Whether $Q_1=Q_2$ holds, doesn't influence the outcome.

Answer 2

You are correct indeed we have that

• $A = 33+173k$
• $B = 17+173h$

then

• $2A+B=83+173(2k+h)$

Regarding the hint, it says that we can assume $h=k$ maybe to facilitate an evaluation by direct calculation but obviously it is not a necessary condition.