# HCF and LCM tricks for CAT

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Quantitative Aptitude is one of the three sections in the CAT. Practice is the key to improve in the Quantitative Aptitude section. Hence check out a few free mocks for CAT here. Consequently, ensure that you analyze the solutions after taking the mock. If we look at Previous Question papers for CAT for HCF and LCM CAT Questions, we can see that questions on LCM and HCF are quite commonly asked in CAT. The theory behind HCF and LCM tricks for CAT questions is very straight forward and hence this is an area where students can easily score marks if they know the theory behind the concept.

# HCF and LCM tricks for CAT

### How to find LCM?

To find the LCM or HCF of a set of numbers, first represent them as the product of prime factors. While calculating LCM, find the highest exponent for each prime factor among the numbers. The LCM is the product of the prime factors raised to the highest exponent.

For example to find the LCM of 720, 140 and 64, we factorize each number as shown below:

$720 = 2^4*3^2*5$

$140 = 2^2*5*7$

$64 = 2^6$

Hence, the highest exponent of each prime factor is as follows:

2 is 6, 3 is 2, 5 is 1 and 7 is 1

Hence LCM = $2^6*3^2*5*7 = 20160$

### How to find HCF?

While calculating HCF, find the lowest exponent for each prime factor among the numbers. The HCF is the product of the prime factors raised to the lowest exponent.

For example to find the HCF of 720, 140 and 64, we factorize each number as shown below:

$720 = 2^4*3^2*5$

$140 = 2^2*5*7$

$64 = 2^6$

Hence for the 3 numbers the lowest exponent are as follows:

2 is 2, 3 is 0, 5 is 0 and 7 is 0.

Hence the HCF $= 2^2 = 4$.

## Properties of HCF and LCM tricks for CAT

1) HCF * LCM of two numbers = Product of two numbers

Hence, if the two numbers are 180 and 320, the HCF=20 and LCM = (180*320)/20= 2880.

2) Whenever the HCF of a set of numbers is given to be a number ‘h’, the numbers can be written of the form ah, bh, ch, dh … where a, b, c, d do not share any common factor.

3) LCM of two co-prime numbers is equal to their product and their HCF is 1.

4) LCM of fractions $\frac{a}{b}$ and $\frac{c}{d}$ is $\frac{LCM(a,c)}{HCF(b,d)}$

5) HCF of fractions $\frac{a}{b}$ and $\frac{c}{d}$ is $\frac{HCF(a,c)}{LCM(b,d)}$

## Applications of HCF and LCM tricks for CAT

1) The greatest number dividing a, b and c leaving remainders of x1, x2 and x3 is the HCF of (a-x1x1), (b-x2x2) and (c-x3x3).

For example if a number divides 51, 128, 298 and leaves 3, 8 and 10 as the remainders then the largest number to do so would be the HCF of (48, 120, 288) = 24.

2) The greatest number dividing a, b and c (a<b<c) leaving the same remainder each time is the HCF of (c-b), (c-a), (b-a).

For example the largest number to divide 102, 141 and 258 is the HCF of (258-102, 258-141, 141-102) = HCF of ( 156, 117, 39) = 39.

3) If a number, N, is divisible by X and Y and HCF(X,Y) = 1. Then, N is divisible by X*Y

## Solved Examples using HCF and LCM tricks for CAT

### Question 1

Ajay, Vijay and Alay run on a circular track of 800m at 0.4m/s, 0.5m/s and 0.8m/s. If they start at the same time, how many times will they meet by the time Alay finishes 10 rounds of the track?

#### Solution

The time taken to finish one round of the track for Ajay, Vijay and Alay is 2000s, 1600s and 1000s respectively.

Hence, they meet every x seconds where x=LCM(2000, 1600, 1000) = 8000s.

Hence, by the time Alay completes 10 rounds in 10000s, they would have met exactly once.

### Question 2

A number x leaves the same remainder when it divides 20, 50 and 62. Find the maximum possible value of x?

#### Solution

The greatest number to leave the same remainder is

HCF of (62-20, 62-50, 50-20) = HCF of (42, 12, 30) = 6.

Hence max value of x=6.

### Question 3

The HCF of 24 and a is 8 and the LCM is 48. Find a?

#### Solution

HCF*LCM = 24*a = 8*48.

Hence, a=16.

In conclusion, with the information presented in this post we hope you can now easily solve questions on LCM and HCF for CAT.

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