Common Factors & HCF

overview

This page provides a brief overview of

*common factors of two or more numbers*,

*co-prime numbers*,

*highest common factor of two or more numbers*,

*highest common factor using factorization method*,

*highest common factor using division method*

Common Factors

Factors of 1212 are 1,2,3,4,6,121,2,3,4,6,12 and factors of 1818 are 1,2,3,6,181,2,3,6,18. It is noted that "1,2,3,61,2,3,6" are the common between these two lists.

The word "common" means "one sharing and have a part of the other".

**Common Factors** : Factors common between two of more numbers are common factors of those numbers.

The common factors of 22 and 4 is "2".

The numbers 8 and 9 do not have any common factors, other than 1.

What are the common factors of 8 and 12?

The answer is "2,4".

The factors of 8 are 1,2,4,8.

The factors of 12 are 1,2,3,4,6,12.

The common numbers between the two list is 2 and 4.

Co-Prime Numbers

Factors of 10 are 1,2,5,10

Factors of 21 are 1, 3, 7, 21.

Factors of 6 are 1, 2, 3, 6.

Comparing the factors of 10 and 21, we find that they do not have any common factors, other than 1.

Comparing factors of 10 and 6, we find that they have a common factor 2.

10 and 21 are called "**co-prime numbers**".

10 and 6 are not co-prime numbers, as they share 2 as the common-factor.

Similarly, 6 and 21 are not co-prime numbers, as they share 3 as the common-factor.

**Co-prime Numbers** : Two or more numbers without any common factor, other than 1, are called co-prime numbers.

The prefix "co"? means "together and mutually".

Are 7 and 7 co-prime numbers?

The answer is "no". They have a common factor 7.

Are 21 and 10 co-prime numbers?

The answer is "Yes, they do not have a common factor". The factors of 21 are 1,3,7,21 and 10 are 1,2,5,10.

Highest Common Factor

Factors of 12 are 1,2,3,4,6,12 and factors of 18 are 1,2,3,6,18.

The common factors between these two are "1,2,3,6"

The highest common factor for these two numbers is "6"

**Highest Common Factor** : The highest factor in the common factors of two or more numbers is the highest common factor of those numbers.

What is the HCF of 2 and 4?

The answer is "2"

What is the HCF of 8 and 9?

The answer is "1".

What is the HCF of 24 and 30?

The answer is "6".

The factors of 24 are 1,2,3,4,6,8,12,24.

The factors of 30 are 1,2,3,5,6,10,15,30.

The common factors are 1,2,3,6.

The highest value in common factors is 6.

HCF Factorization Method

Consider finding the HCF of 36 and 48.

36 has 9 factors and 48 has 10 factors. It is not easy to list the factors for larger numbers.

To simplify the problem "prime factorization of numbers" is used.

Consider finding the HCF of 36 and 48.

Prime factorization of 36=2×2×3×3

Prime factorization of 48=2×2×2×2×3

"2×2×3" is common between these two.

Consider finding the HCF of 36 and 48. A simplified procedure is given in the figure. The numbers are divided successively by common factors. The common factors are multiplied to get HCF.

**Factorization Method to find HCF** : The numbers are successively divided by common factors. Product of common factors is the HCF.

An example with numbers 36 and 48 are given in the figure above.

Find the HCF of 280 and 420

The answer is "140".

The product of common factors is 5×2×2×7.
Using the Factorization Method to find the HCF.

HCF Division Method

Consider two numbers with HCF n. The numbers can be given in the form n×p and n×q.

For example, HCF 36 and 48 is 12. And 36=12×3 and 48=12×4.
In this example n=12, p=3 and q=4.

The p and q are derived from non-common prime factors of the two numbers.

Let us consider that n×q is the larger than n×p.

To simplify finding HCF, the larger number is subtracted by a multiple of the smaller number. That is, n×q-m×n×p is the difference, where m is randomly chosen.

The HCF of the two numbers n×p and n×q-m×n×q is "same as the HCF of n×p and n×q".

It was learned that HCF of two numbers n×p and n×q is same as the HCF of n×p and n×q-m×n×p for any m. Using this property, a simplified procedure is devised.

The simplified procedure is given in the figure.

• The numbers are placed in long division form

• A multiple of smaller number is subtracted from the larger number (420-280).

• Now, the difference and the smaller number are the pair for which the HCF is to be found.
repeat the procedure in the next step. The step in which the difference is 0, the smaller number in that step is the HCF.

**Division Method to find HCF** : To find HCF of two large numbers :

• The numbers are placed in long division form

• A multiple of smaller number is subtracted from the larger number.

• Now, the difference and the smaller number are the pair for which the HCF is to be found.
repeat the procedure in the next step. The step in which the difference is 0, the smaller number in that step is the HCF.

Find HCF of 336 and 72. The answer is "24".

summary

**Common Factors** : Factors common between two of more numbers are common factors of those numbers.

**Co-prime Numbers** : Two or more numbers without any common factor, other than 1, are called co-prime numbers.

**Highest Common Factor** : The highest factor in the common factors of two or more numbers is the highest common factor of those numbers.

**Factorization Method to find HCF** : The numbers are successively divided by common factors. Product of common factors is the HCF.

An example with numbers 36 and 48 are given in the figure.

**Division Method to find HCF** : To find HCF of two large numbers :

• The numbers are placed in long division form

• A multiple of smaller number is subtracted from the larger number.

• Now, the difference and the smaller number are the pair for which the HCF is to be found.
repeat the procedure in the next step. The step in which the difference is 0, the smaller number in that step is the HCF.

Outline

The outline of material to learn "Divisibility in Whole Numbers" is as follows.

Note: * click here for detailed outline of Whole divisors *

→ __Classification as odd, even, prime, and composite__

→ __Factors, Multiples, Prime factorization__

→ __Highest Common Factor__

→ __Lowest Common Multiple__

→ __Introduction to divisibility tests__

→ __Simple Divisibility Tests__

→ __Simplification of Divisibility Tests__

→ __Simplification in Digits for Divisibility Tests__