The GRE maths sample question given below is a problem solving practice question from the topic number properties. It tests the concept of HCF of two numbers and coprime numbers.

Question 2 : How many pairs of natural numbers whose HCF is 12 add up to 216?

- 6
- 3
- 9
- 18
- 17

From INR 2999

The sum of the pair of natural numbers = 216

The HCF of the pair of natural numbers = 12

HCF of the two numbers is 12. So, both numbers are divisible by 12.

In other words, both numbers are multiples of 12.

Let the two numbers be 12x and 12y, where x and y are coprime.

Given, 12x + 12y = 216

x + y = 18

**Note**: Two numbers a and b are said to be co-prime if their highest common factor(HCF) is 1

i.e., any prime that divides one number will not divide the other number.

**x and y can take the following values:**

X | Y |
---|---|

1 | 17 |

5 | 13 |

7 | 11 |

The pairs of natural numbers are in the form (12x, 12y)

Therefore, pairs of natural numbers whose HCF is 12 add up to 216 are **(12, 204), (60, 156), (84, 132)**

Number of pairs of natural numbers whose HCF is 12 add up to 216 = **3**

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