GRE® Quant Practice Q8

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?

1. 243
2. 144
3. 240
4. 112
5. 560
6. 405

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Video Explanation

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Explanatory Answer | GRE Problem Solving Question 8

Approach: When does a number completely divide another number?

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.

Step 1: Prime factorise 5040

The prime factorised form of 5040 is 2⁴ × 3² × 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.

Step 2: Prime factorise the answer options

Choice A: 243
The prime factorised form of 243 is 3⁵.
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 2⁴ × 3².
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 2⁴ × 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 2⁴ × 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 2⁴ × 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 3⁴ × 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.

Choices B, C, D, E are the correct answers.