We will discuss here about the method of h.c.f. (highest common factor).
Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D)
The highest common factor or HCF of two or more numbers is the greatest number which divides exactly the given numbers.
1. Let us consider two numbers 16 and 24.
Factor of 16 are → 1, 2, 4, 8, 16
Factor of 24 are → 1, 2, 3, 4, 6, 8, 12, 24 |
1 × 16, 2 × 8, 4 × 4 1 × 24, 2 × 12, 3 × 8, 4 × 6 |
We see that the highest common factor of 16 and 24 is 8. In short, the Highest Common Factor is expressed as H.C.F.
2. Find the H.C.F. of 12 and 18.
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors of 12 and 18 = 1, 2, 3, 6
Highest common factor (H.C.F) of 12 and 18 = 6
3. Find the H.C.F. of 15 and 28.
Factors of 15 = 1, 3, 5, 15
Factors of 28 = 1, 2, 4, 7, 14, 28
Common factors of 15 and 28 = 1
Highest common factor (H.C.F) of 15 and 28 = 1
Two numbers which have only 1 as the common factor are called co-prime.
Finding H.C.F.
There are three methods of finding H.C.F. of two or more numbers.
1. Factorization Method
2. Prime Factorization Method
3. Division Method
1. H.C.F. by Factorization Method:
Let us consider some examples.
I. Find the H.C.F. of 36 and 45.
Factor of 36 are → 1, 2, 3, 4, 6, 9, 12, 18, 36
Factor of 45 are → 1, 3, 5, 9, 15, 45 |
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 1 × 45, 3 × 15, 5 × 9 |
The common factors of 36 and 45 are 1, 3, 9.
The highest common factor is 9.
II. Find the HCF of 12, 48 and 72.
Let us first list all the factors of each number.
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48
Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72
The common factors of 12, 48 and 7 are 1, 2, 3, 4, 6 and 12.
The highest common factor is 12.
2. H.C.F. by Prime Factorization Method
Let us consider an example.
I. Find the H.C.F. of 24, 36 and 48.
First we find the prime factors of 24, 36 and 48.
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
The common prime factors = 2, 2, 3
H.C.F. = 2 × 2 × 3 = 12
II. Find the H.C.F. of 132 and 330.
2 | 132 2 | 66 3 | 33 11 |
132 ÷ 2 = 66 66 ÷ 2 = 33 33 ÷ 3 = 11 |
2 | 330 5 | 165 3 | 33 11 |
330 ÷ 2 = 165 165 ÷ 5 = 33 33 ÷ 3 = 11 |
132 = 2 × 2 × 3 × 11
330 = 2 × 3 × 5 × 11
The common factors are 2, 3, 11
Therefore, H.C. F. = 2 × 3 × 11
= 66
Here 66 is also the greatest common divisor of 132 and 330.
132 ÷ 66 = 2;
330 ÷ 66 = 5
IV: Find the H.C.F. of 128 and 160. 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 160 = 2 × 2 × 2 × 2 × 2 × 5 The common factors are 2, 2, 2, 2, 2 H.C.F. = 2 × 2 × 2 × 2 × 2 = 32 |
2 | 128 2 | 64 2 | 32 2 | 16 2 | 8 2 | 4 2 |
2 | 160 2 | 80 2 | 40 2 | 20 2 | 10 5 |
3. H.C.F. by Division Method
This method can also be used to find the H.C.F. of more than 2 numbers.
Let us consider a few examples.
1. Find the H.C.F. of 12 and 18.
Step I: Treat the smallest number i.e., 12 as divisor and the bigger number i.e., 18 as dividend. Step II: The remainder 6 becomes the divisor and the divisor 12 becomes the dividend. Step III: Repeat this process till the remainder becomes zero. The last divisor is the H.C.F. |
2. Find the H.C.F. of 16, 18 and 24.
Step I: First we consider the first two numbers and follow the same step 1, 2 and 3 of the above example. Step II: The H.C.F. of the first two numbers which is 2 becomes the divisor and the third number 24 becomes the dividend. This process is repeated till the remainder becomes 0. H.C.F. is the last divisor. |
3. Find the HCF of 18 and 54 by short division method.
Solution:
Write the number in a row separated by commas, divide the numbers by common prime factors. Factorisation stops when we reach prime numbers which cannot be further divided.
HCF is the product of all the common factors.
Hence, the common factors are 2, 3 and 3.
HCF of 18 and 54 = 2 × 3 × 3 = 18.
4. Find the HCF of 28 and 36 by short division method.
Solution:
First we need to write the number in a row separated by commas, divide the numbers by common prime factors. Factorisation stops when we reach prime numbers which cannot be further divided.
HCF is the product of all the common factors.
Hence, the common factors are 2, 2.
HCF of 28 and 36 = 2 × 2 = 4.
5. Find the H.C.F. of 48 and 90.
Steps Divide 48 and 90 by 2. 48 ÷ 2 = 24; 90 ÷ 2 = 45 Divide 24 and 45 by 3. 24 ÷ 3 = 8; 45 ÷ 3 = 15 8 and 15 do not have a common factor. Stop the division. |
Common factors are 2, 3
Therefore, H.C.F of 48 and 90 = 2 × 3 = 6
6. Find the H.C.F. of 36, 54 and 72.
Steps Divide 36, 54 and 72 by 2. 36 ÷ 2 = 18; 54 ÷ 2 = 27; 72 ÷ 2 = 36 Divide 18, 27 and 36 by 3. 18 ÷ 3 = 6; 27 ÷ 3 = 9; 36 ÷ 3 = 12 Divide 6, 9 and 12 by 3. 6 ÷ 3 = 2; 9 ÷ 3 = 3; 12 ÷ 3 = 4 2, 3 and 4 do not have a common factor. Stop the division. |
Common factors are 2, 3, 3
Therefore, H.C.F of 36, 54 and 72 = 2 × 3 × 3 = 18
Questions and Answers on Method of H.C.F.:
I. Find the H.C.F. of the following by prime factorisation method.
(i) 44, 66
(ii) 6, 18
(iii) 675, 1125
(iv) 42, 63
(v) 81, 144
(vi) 78, 104
(vii) 64, 48
(viii) 96, 80
(ix) 24, 48
(x) 200, 400
Answer:
I. (i) 22
(ii) 6
(iii) 225
(iv) 21
(v) 9
(vi) 26
(vii) 16
(viii) 16
(ix) 24
(x) 200
II. Find the H.C.F. of the following by division method.
(i) 112, 256
(ii) 25, 65
(iii) 24, 36, 48
(iv) 7, 21, 35
(v) 30, 45, 75
(vi) 60, 75
(vii) 12, 24, 36
(viii) 90, 128, 144
(ix) 55, 88
(x) 24, 56
Answer
II. (i) 16
(ii) 5
(iii) 12
(iv) 7
(v) 15
(vi) 15
(vii) 12
(viii) 2
(ix) 11
(x) 8
III. Find the H.C.F. of the following by writing all the factors.
(i) 16, 18
(ii) 12, 36, 9
(iii) 75, 80
(iv) 6, 12, 15
(v) 20, 30, 60
Answer:
III. (i) 2
(ii) 3
(iii) 5
(iv) 3
(v) 10
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