This GRE maths sample question is a Select One or More Answers question in Number Properties. Concept tested: Divisibility of a number.

Question 8: Which of the following numbers will divide 5040 completely without leaving a remainder? Indicate __all__ such numbers?

- 243
- 144
- 240
- 112
- 560
- 405

From INR 2999

A number will completely divide another number when the all the prime factors of the divisor are present in the dividend.

If the dividend does not have one of the prime factors of the divisor or does not have it in adequate powers, then the divisor will not divide the dividend completely.

The prime factorised form of 5040 is 2^{4} × 3^{2} × 5 × 7. Hence, the answers will have only the factors among 2, 3, 5 and 7, with their highest powers being less than or equal to 4, 2, 1 and 1, respectively.

**Choice A: 243**

The prime factorised form of 243 is 3^{5}.

The highest power of 3 in 5040 is 2.

Hence, 243 will not divide 5040 and is __not__ a correct answer.

**Choice B: 144**

The prime factorised form of 144 is 2^{4} × 3^{2}.

The highest power of 2 in 5040 is 4.

The highest power of 3 in 5040 is 2.

Hence, 144 will divide 5040 and is a correct answer.

**Choice C: 240**

The prime factorised form of 240 is 2^{4} × 3 × 5.

The highest power of 2 in 5040 is 4.

The highest power of 3 in 5040 is 2.

The highest power of 5 in 5040 is 1.

Hence, 240 will divide 5040 and is a correct answer.

**Choice D: 112**

The prime factorised form of 112 is 2^{4} × 7.

The highest power of 2 in 5040 is 4.

The highest power of 7 in 5040 is 1.

Hence, 112 will divide 5040 and is a correct answer.

**Choice E: 560**

The prime factorised form of 560 is 2^{4} × 5 × 7.

The highest power of 2 in 5040 is 4.

The highest power of 5 in 5040 is 1.

The highest power of 7 in 5040 is 1.

Hence, 560 will divide 5040 and is a correct answer.

**Choice F: 405**

The prime factorised form of 405 is 3^{4} × 5.

The highest power of 3 in 5040 is 2. The divisor has more 3s than the dividend.

The highest power of 5 in 5040 is 1.

Hence, 405 will not divide 5040 and is __not__ a correct answer.

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