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Number Theory : Remainders, Divisors

Remainders of division of two different numbers and their sum by the same divisor

Question

When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?

(1) 11
(2) 17
(3) 13
(4) 23

Correct Choice is (3) and Correct Answer is 13

Explanatory Answer

Let the divisor be d.

When 242 is divided by the divisor, let the quotient be 'x' and we know that the remainder is 8.
Therefore, 242 = xd + 8

Similarly, let y be the quotient when 698 is divided by d.
Then, 698 = yd + 9.

242 + 698 = 940 = xd + yd + 8 + 9
940 = xd + yd + 17
As xd and yd are divisible by d, the remainder when 940 is divided by d should have been 17.

However, as the question states that the remainder is 4, it would be possible only when leaves a remainder of 4.

If the remainder obtained is 4 when 17 is divided by d, then d has to be 13.

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