Short Tricks to solve HCF with Solved Example

The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), is the biggest number that can evenly divide two or more numbers. For example, for the numbers 12 and 15, the HCF is 3 because 3 is the largest number that divides both 12 and 15 without leaving any remainder.

In this article, we'll show different short tricks to calculate the HCF of the given numbers and provide examples and practice problems.

Short Tricks to Solve HCF

One of the short tricks to find the HCF of any two numbers is called repeated subtraction. Let's discuss this method in detail.
Euclidean Algorithm (Subtraction Trick)
Subtract the smaller number from the larger one repeatedly until both numbers are the same. This number is the HCF.

Let's consider an example for better understanding.
Example: Find HCF for 56 and 98.

Solution:

• 98 - 56 = 42
• 56 - 42 = 14
• 42 - 14 = 28
• 28 - 14 = 14 (repeated until both are 14)
• HCF: 14

Some Other Tips and Tricks Related to HCF

The greatest number that divides with a remainder:

• When a number divides a, b, and c, leaving a remainder R, the HCF of the numbers is obtained by subtracting the remainder from each number.

Greatest number that divides with a remainder(R) = HCF(a − R, b − R, c − R)

• Greatest number dividing with different remainders: To find the greatest number that divides x, y, and z, leaving remainders a, b, and c respectively, the greatest number is the HCF of x - a, y − b, and z − c.

• HCF of Two Numbers Using LCM: The product of two numbers is equal to the product of their HCF and LCM.

a × b = HCF(a, b) × LCM(a, b)

• HCF of any two consecutive numbers (like 8 and 9 or 99 and 100) is always 1. This is because consecutive numbers share no common factors other than 1.

• The H.C.F. of two or more numbers greater than zero is always smaller than or equal to the smallest of given numbers.

Example: HCF ( 24, 36, 40) <= 24.

• If one number is a factor of another, the smaller number is the HCF of those two numbers.

Example: HCF(16, 32) = 16 as 16 divides 32.

• The HCF of fractions is equal to the HCF of numerators of the fractions divided by the LCM of Denominators of the fractions.

\text{HCF of Fractions} = \frac{HCF\ of\ Numerators}{LCM\ of\ Denominators}

• The HCF of Prime numbers is always 1 as they do not have any common factor other than 1.

Solved Examples of short Tricks to Find HCF

Example 1: Find the HCF of 84 and 126.

Solution:

Step 1: Subtract the smaller number (84) from the larger number (126) i.e., 126 − 84 = 42
Step 2: Now, subtract the result (42) from the smaller number (84) i.e., 84 − 42 = 42
Step 3: Since both numbers are now equal (42), we stop here.

The HCF of 84 and 126 is 42.

Example 2: Find the greatest number that divides 38, 50, and 74 leaving a remainder of 2 in each case.

Solution:

Given Values:

• Numbers: a = 38, b = 50, c = 74
• Remainder: R = 2

Adjust the Numbers by Subtracting the Remainder:

• a − R = 38 − 2 = 36
• b − R = 50 − 2 = 48
• c − R = 74 − 2 = 72

Find the HCF of 36, 48, and 72:

• Prime factorization of 36: 36 = 2² × 3²
• Prime factorization of 48: 48 = 2⁴ × 3
• Prime factorization of 72: 72 = 2³ × 3²

The common prime factors are 2 and 3.

• The lowest power of 2 common to all three numbers is 2².
• The lowest power of 3 common to all three numbers is 3.

Therefore, the HCF is:HCF(36, 48, 72) = 2² × 3 = 4 × 3 = 12

Example 3: Given that the LCM of 15 and 20 is 60, find the HCF of 15 and 20.

Solution:

Given:

• a=15
• b=20

LCM(15, 20) = 60
a × b = HCF(a, b) × LCM(a, b)
15 × 20 = HCF(15, 20) × 60
300 = HCF(15, 20) × 60

HCF(15, 20) = 300/60 = 5

The HCF of 15 and 20 is 5.

Example 4:Find the HCF of \frac{4}{9}\ ​ and\ \frac{6}{15}

Solution :

HCF of fractions = HCF of Numerators / LCM of Denominators.

HCF of \frac{4}{9}\ ​ and\ \frac{6}{15} = \frac{ HCF (4,6)} {LCM ( 9,15)}

then,
HCF ( 4, 6) = 2
LCM ( 9, 15) = 45
HCF of \frac{4}{9}\ ​ and\ \frac{6}{15} = 2/45.

Read More,

• HCF and LCM
• Factorization
• Prime Factorization