# Number Properties DS - Remainders, divisors

Quite a lot of interesting DS questions appear from Number Properties and Number Theory. Here is one such question.

When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?
1. When 2x is divided by d, the remainder is 23.
2. When 3x is divided by d, the remainder is 22.

Choice A. Statement 1 alone is sufficient; statement 2 is not sufficient.

Statement 1:When 2x is divided by d, the remainder is 23.

The question stem states that when x is divided by d, the remainder is 24.

Therefore, when 2x is divided by d, the remainder should be 2*24 = 48.

However, if 48 is greater than or equal to the divisor, the effective remainder is obtained by subtracting a multiple of the divisor from 48 to make the final remainder less than the divisor.

From statement 1, we know that the effective remainder is 23.

So, we can infer that 48 was greater than the divisor and hence a multiple of 'd' was subtracted from 48.

i.e., 48 - nd = 23 or nd = 25.

The possible values for d are 1, 5 and 25.
However, as the remainder when x is divided by d is 24, the divisor cannot be 1 or 5.
So, we can conclude that 25 is the divisor.

Statement 1 is sufficient to find the answer.
Correct answer is either Choice A or Choice D.

Statement 2: When 3x is divided by d, the remainder is 22.

If x leaves a remainder of 24 when divided by d, then 3x will leave a remainder of 3*24 = 72 when divided by d.

However, if 72 is greater than or equal to the divisor, we will subtract a multiple of the divisor to make the effective remainder less than the divisor.

Note : For any division of a number by a divisor d, remainders will take values from 0 to (d -1).

From statement 2, when 3x is divided by d, the remainder is 22.
So, 22 = 72 - n*d
Or nd = 72 - 22 = 50.
nd = 50, d could either be 50 or 25 or 10 or 5 or 2.

However, as the remainder when x is divided by d is 24, d cannot be less than 24.
So, d could either be 25 or 50.

From statement 2 we are not able to deduce a unique value for d.

Hence, statement 2 is not sufficient.

Choice A is the corrrect answer