LCM and HCF basic formulas for the CAT Exam

LCM and HCF basic formulas for the CAT Exam

CAT exam LCM and HCF is one of the important topics for CAT preparation. It is part of number system questions. Number system questions contain 20% weightage in the CAT exam. However, no direct questions will be asked from this topic in CAT. Question Paper but the basics will help you in solving number system questions which have 20% weightage.

LCM and HCF basic concept

HCF and LCM is another very important topic from the number system. The concept is not just restricted to the number system but is also helpful in solving some questions from arithmetic which is a very important topic for CAT exam.

In this article, we will take the topic from basics and will try to understand the various types of problems which require from this concepts. Many questions from this topic in CAT exam are solved by applying direct formulas and tricks. As and when we come across such questions, we will also take note of those formulas.

Methods to find HCF and LCM - Important formula and tricks

HCF or Highest Common Factor is considered as the most known greatest common divisor of two or more positive integers that are the reasons for the division of the numbers without leaving a remainder.

LCM or Least Common Multiple can be defined as the least number that is divisible by two or more than two numbers. For example, 4 and 6 are taken as two numbers.

HCF Prime Factorisation Method

To explain HCF Prime Factorisation Method, the HCF of 144, 104 and 160 are given with solution

The prime factors of 144, 104 and 160.
144 = 2 × 2 × 2 × 2 × 3 × 3
104 = 2 × 2 × 2 × 13
160 = 2 × 2 × 2 × 2 × 2 × 5
The common factors of 144, 104 and 160 are 2 × 2 × 2 = 8
So, HCF (144, 104, 160) = 8

HCF Division Method

To explain the HCF Division Method, the HCF of 144 and 160 are mentioned.
As 160>144, therefore the dividend = 160 and the divisor = 144.
Here, 16 is the highest number that divides 160 and 144.
So, HCF (144, 160) = 16

LCM Prime Factorisation Method

To explain LCM Prime Factorisation Method, the LCM of 60 and 45 are given with solution

Prime Factorisation of 60 and 45
60 = 2 × 2 x 3 × 5
45 = 3 × 3 × 5
LCM = 2 × 2 x 3 × 3 × 5 = 180

LCM Division Method

To explain LCM Division Method, the LCM of 60 and 45 are given with solution

Division of 60 and 45
LCM of 60 and 45 = 2 × 2 x 3 × 3 × 5 = 180

Video’s that could help aspirant.

CAT Quantitative Aptitude: HCF and LCM Important Formulas

In the section below, formulas regarding HCF and LCM are mentioned.

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

If A and B are taken as the numbers the formula will be A x B = H.C.F.(A,B) x L.C.M.(A,B)

List of Properties

HCF and LCM Property 1

The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers. LCM × HCF = Product of the Numbers. Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

For Example: If 3 and 8 are two numbers.

LCM (3,8) = 24

HCF (3,8) = 1

LCM (3,8) x HCF (3,8) = 24 x 1 = 24

Also, 3 x 8 = 24

Hence, proved.

Note: This property is applicable for only two numbers.

HCF and LCM Property 2

HCF of co-prime numbers is 1. Therefore, the LCM of given co-prime numbers is equal to the product of the numbers. LCM of Co-prime Numbers = Product Of The Numbers

For Example: Let us take two coprime numbers, such as 21 and 22.

LCM of 21 and 22 = 462

Product of 21 and 22 = 462

LCM (21, 22) = 21 x 22

HCF and LCM Property 3

H.C.F. and L.C.M. of Fractions:

LCM of fractions = LCM of Numerators / HCF of Denominators. HCF of fractions = HCF of Numerators / LCM of Denominators. For Example: Let us take two fractions 4/9 and 6/21. 4 and 6 are the numerators & 9 and 12 are the denominators

LCM (4, 6) = 12

HCF (4, 6) = 2

LCM (9, 21) = 63

HCF (9, 21) = 3

Now as per the formula, we can write:

LCM (4/9, 6/21) = 12/3 = 4. Then HCF (4/9, 6/21) = 2/63

HCF and LCM Property 4

HCF of any two or more numbers is never greater than any of the given numbers. For Example: HCF of 4 and 8 is 4. Here, one number is 4 itself and another number 8 is greater than HCF (4, 8), i.e.,4.

HCF and LCM Property 5

LCM of any two or more numbers is never smaller than any of the given numbers. For Example: LCM of 4 and 8 is 8 which is not smaller to any of them

Solved Problems

For Example 1: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12

Solution:
9 = 3 × 3 = 3²
12 = 2 × 2 × 3 = 2² × 3
LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 9 × 12 = 108. Proved.

For Example 2: 8 and 9 are two co-prime numbers. Using these numbers verify, LCM of Co-prime Numbers = Product Of The Numbers.

Solution: LCM and HCF of 8 and 9:

8 = 2 × 2 × 2 = 2³

9 = 3 × 3 = 3²

LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72

HCF of 8 and 9 = 1

Product of 8 and 9 = 8 × 9 = 72

Hence, LCM of co-prime numbers = Product of the numbers. Therefore, verified.

CAT Preparation Tips

Some tips for the preparation for CAT HCF and LCM are mentioned below.

Develop a clear idea on the topics.
Revise the formulas by practicing daily.
Do solve previous years question paper
Must give a mock test to increase speed for the exam.
Start from the basic questions like
- Find the Highest Common Factor of 25, 35 and 45.
- Find the Least Common Multiple of 36 and 44.

Questions for Practice

Question 1 Find the smallest number that leaves a remainder of 4 on division by 5, 5 on division by 6, 6 on division by 7, 7 on division by 8 and 8 on division by 9?

2519
5039
1079
979

Question 2: There are three numbers a,b, c such that HCF (a, b) = l, HCF (b, c) = m and HCF (c, a) = n. HCF (l, m) = HCF (l, n) = HCF (n, m) = 1. Find LCM of a, b, c. (The answer can be "This cannot be determined").

Question 3: In a large school auditorium. The students are made to sit to watch the programmes. If the teachers make a row of students of 16 each, there will be 12 students left. If they make rows of 24 each, then there will be 20 students left. So If they make rows of 25 each. There will be 21 students left and if they make rows of 30 each, there will be 26 students left. What is the minimum number of students present in the school?