# CAT Practice : Number System - HCF, LCM

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If we " take away" the HCF from two numbers, the pair of numbers we are left with have to be coprime. This is a fabulous starting point for a number of questions.

## HCF basics

Q.4: How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?
1. 12
2. 8
3. 15
4. 30

Choice (C). 15 Pairs

## Detailed Solution

Let HCF of (x, y) be h. Then we can write x = h * a and y = h * b. Furthermore, note that HCF (a, b) = 1. This is a very important property. One that seems obvious when it is mentioned but a property a number of people overlook.

So, we can write x = 35a; y = 35b

x + y = 1085 => 35(a + b) = 1085. => (a + b) = 31. We need to find pairs of co-prime integers that add up to 31. (Another way of looking at it is to find out integers less than 31 those are co-prime with it or phi(31) as had mentioned. More on this wonderful function in another post).

Since 31 is prime. All pairs of integers that add up to 31 will be co-prime to each other. Or, there are totally 15 pairs that satisfy this condition.
Choice (C).

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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.