Number Properties - divisibility and remainders

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.

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