This GRE maths sample question is a problem solving question in Number Properties. Concept tested: Euclid's Division Algorithm & Properties of Remainders.

Question 6: If n is a positive integer that leaves a remainder of 45 when divided by 60, which of the following __cannot__ be a divisor of n?

- 9
- 15
- 45
- 30
- 25

From INR 2999

**Given**: n is a positive integer that leaves a remainder of 45 when divided by 60.

Let the quotient when n is divided by 60 be q. n can be represented as

n = 60q + 45

Since the quotient, q, is multiplied by 60, it will definitely be an even number.

45 is an odd number.

The sum of an even and an odd number will definitely be an odd number.

An odd number will never be divisible by an even number.

Hence, the divisor can never be an even number.

Among the answer options, only option D, which is 30, can never be a divisor of n.

Hence, Choice D is the correct answer.

**Choice A: 9**

n = 60q + 45

45 is divisible by 9.

If the term 60q is divisble by 9, then n will be divisible 9.

Take q to be equal to 9, n will be divisible by 9.

We have got one scenario where n is divisible by 9.

9 can be a divisor and hence, is not the answer.

**Choice B: 15**

n = 60q + 45

45 is divisible by 15.

60 is also divisible by 15.

Hence, any integral value of q will make n divisible by 15.

15 will be a divisor and hence, is not the answer.

**Choice C: 45**

n = 60q + 45

45 is divisible by 45.

If the term 60q is divisble by 45, then n will be divisible 45.

Take q to be equal to 45, n will be divisible by 45.

We have got one scenario where n is divisible by 45.

45 can be a divisor and hence, is not the answer.

**Choice D: 30**

n = 60q + 45

60 is divisible by 30.

45 is not divisible by 30.

Hence, the sum 60q + 45 will not be divisible by 30

45 cannot be a divisor and is the answer.

**Choice E: 25**

n = 60q + 45

Take q to be equal to 3.

n = 60 × 3 + 45 = 225.

225 is divisible by 25 We have got one scenario where n is divisible by 25.

25 can be a divisor and hence, is not the answer.

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