When a positive integer A is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. Which of the following is a factor of (A – B)?
(A) 12
(B) 24
(C) 35
(D) 16
(E) 30
Correct Answer : Choice C
Explanatory Answer
When A is divided by 5, the remainder is 3. So, we can express A = 5x + 3
When B is divided by 5, the remainder is 3. So, we can express B = 5y + 3
So, (A – B) = 5x + 3 – (5y + 3) = 5(x – y). So, (A – B) is a multiple of 5.
Similarly, when A is divided by 7, the remainder is 4. So, we can express A = 7p + 4
When B is divided by 7, the remainder is 4. So, we can express B = 7q + 4
So, (A – B) = 7p + 4 – (7q + 4) = 7(p – q). So, (A – B) is a multiple of 7.
Combining the two results, we can conclude that (A – B) is a multiple of both 5 and 7.
i.e., (A – B) will be a multiple of 35.