The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x +7. Which one of the following could m equal?

Question: The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x +7. Which one of the following could m equal?

Solution:
Given that
The remainder for m/14 = x+7 [since remainder must be between 0 and 13]
Also,
The remainder for m/7 = x, [since remainder must be between 0 and 6]
Check options:
when m is divided by 7 remainder is x but when x is divided by 14 the remainder becomes x+7.

ক) 45/7 = (6 × 7) + 3
but 45/14 = (14 × 3) + 3
So, 45 is eliminated as remainder is same in both cases.

খ) 53/7=(7 × 7) + 4
53/14=(14 × 3) + 11.
Look at the remainder here. Remainder is 11. we know x = 4 when m is divided by 7. Now we have remainder
x+7 = 4 + 7 = 11
So, 53 is the correct answer.

গ) 72/7 = (7 × 10) + 2
72/14 = (14 × 5) + 2
So, 72 is eliminated as remainder is same in both cases.

ঘ) 85/7 = (7 × 12) + 1
85/14 = (14 × 6) + 1
So, 85 is eliminated as remainder is same in both cases.