The hcf of two numbers is 28 and the hcf of two other numbers is 82. find the hcf of all these four

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We will discuss here about the method of h.c.f. (highest common factor).

The highest common factor or HCF of two or more numbers is the greatest number which divides exactly the given numbers.

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.

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.

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

3. H.C.F. by division method

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.

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