# LCM and HCF Formulas

## Basic Formulas of LCM and HCF with Definition

On this page we have discussed LCM and HCF formulas, definition with examples.

LCM stands for Least Common Factor

LCM or least common factor of two numbers 4, 6 is denoted as LCM(4, 6). And the LCM is the smallest positive integer that is divisible by both 4 and 6, which is 12.

HCF stands for Highest Common Factor

Greated Common Divisor or gcd of two or more positive integers is defined as the largest positive integer that divides the numbers without leaving the remainder. ## LCM and HCF Formulas

### HCF by Prime Factorization Method

Take an example of finding the highest common factor of 100, 125 and 180.
Now let us write the prime factors of 100, 125 and 180.
100 = 2 × 2 × 5 × 5
125 = 5 × 5 × 5
180 = 3 × 3 × 2 × 2 × 5
The common factors of 100, 125 and 180 are 5
Therefore, HCF (100, 125, 180) = 5

### HCF by Division Method

Steps to find the HCF of any given numbers:

1. Larger number/ Smaller Number
2. The divisor of the above step / Remainder
3. The divisor of step 2 / remainder. Keep doing this step till R = 0(Zero).
4. The last step’s divisor will be HCF.

Example:

Let’s take two number 120 and 180

120) 180 (1
120
---------
60) 120 (2
120
---------
000

### LCM by Prime Factorization Method

A technique to find the Least Common Multiple (LCM) of a set of numbers by breaking down each number into its prime factors and then multiplying the highest powers of each prime factor.

Lets take two numbers i.e., 25 and 35, now to calculate the LCM:

• List the prime factors of each number first.
25 = 5 × 5
35 = 7 × 5
• Then multiply each factor the most number of times it occurs in any number.

If the same multiple occurs more than once in both the given numbers, then multiply the factor by the most number of times it occurs.
The occurrence of Numbers in the above example:
5: two times
7: one time
LCM = 7 × 5 × 5 = 175

### LCM by Division Method

Let us see with the same example, which we used to find the LCM using prime factorization.
Solve LCM of (25,35) by division method.

5 | 25, 35----------5 | 5, 7---------7 | 1, 7---------  | 1, 1

Therefore, LCM of 25 and 35 = 5  x 5 × 7 = 175

### Questions and Answers of HCF and LCM

Question:

Calculate the highest number that will divide 43, 91 and 183 and leaves the same remainder in each case

Options

A. 4
B. 7
C. 9
D. 13

Solution:

Here the trick is :

1. Find the Differences between numbers
2. Get the HCF ( that differences)

We have here 43, 91 and 183

So differences are

183 – 91 = 92,
183 – 43 = 140,
91 – 43 = 48.

Now, HCF (48, 92 and 140)

• 48 = 2 × 2 × 2 × 2 × 3
• 92 = 2 × 2 × 23
• 140 = 2 × 2 × 5 × 7
• HCF = 2 × 2 = 4

And 4 is the required number.

Question:

Which of the following is  greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

Options

A. 9700
B. 9600
C. 9800
D. 9650

Solution:    Greatest number of 4-digits is 9999.

Now , find the L.C.M. of 15, 25, 40 and 75 i.e.  600.

On dividing 9999 by 600, the remainder is 399.

Hence,  Required number (9999 – 399) = 9600.

Alternatively,

$\frac{9999}{600}$ = 16.66500

Ignore the decimal points, required number would be 16 * 600 = 9600

Question:

The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is:

Options

A. 25 cm
B. 15 cm
C. 35 cm
D. 55 cm

Solution:     Required length = H.C.F. of 700 cm, 385 cm and 1295 cm = 35 cm.

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### 4 comments on “LCM and HCF Formulas”

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