# CAT Practice : Number System - HCF, LCM

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Given sum of two numbers and their GCD, how do we find the two numbers.

## HCF basics

Q.7: The sum of two non co–prime numbers added to their HCF gives us 91. How many such pairs are possible?
1. 2
2. 4
3. 3
4. 6

Choice C. 3 Pairs

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## Detailed Solution

Let HCF of the numbers be h. The numbers can be taken as ha + hb, where a, b are coprime.
h + ha + hb = 91
h(1 + a + b) = 91
h ≠ 1
h = 7
=> 1 + a + b = 13 a + b = 12

h = 13
=> 1 + a + b = 7
=> a + b = 6

Case 1: h = 7, a + b = 12
(1, 11), (5, 7) => Only 2 pairs are possible as a, b have to be coprime.

Case 2: h = 13, a + b = 6
(1, 5) only one pair is possible as a, b have to be coprime.

Overall, 3 pairs of numbers are possible – (7, 77) (35, 49) and (13, 65)

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## More questions from Number Theory - HCF, LCM

Number Theory is one of the most heavily tested topics. Within this, one should get the basics on factors, multiples, HCF, LCM very clear before moving on to the tougher sets.