Question

Using Euclid's Division Lemma, find HCF of 54 and 90.​

Answer 1

Given : Two numbers 54 * 90  , Euclid's Division Lemma

To Find : HCF of 54 & 90 using Euclid's Division Lemma

Solution:

90 = 54 * 1  + 36

54 = 36 * 1  + 18

36  = 18 * 2  + 0

=> 18 is the HCF  of 54 & 90

                     

54 _|  90    |_1

           54

         _____

             36   _|    54   |_  1

                           36

                         ____

                          18     _|    36   |_2

                                         36

                                        ___

                                           0

18 is the HCF

using Prime factorization

54 = 2 * 3 * 3 * 3

90 = 2 * 3 * 3 * 5

Common Factors , 2 , 3 , 3

HCF = 2 * 3 * 3 = 18

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Answer 2

Given:

Two numbers are 54 and 90.

To find:

The HCF of given numbers using Euclid's Division Lemma.

Solution:

According to Euclid's Division Lemma, if a and b are two positive integers, then there exist unique positive integers q and r, such that

$a=bq+r$

where, q is quotient and r is remainder, $0\leq r<b$.

We know that, 90>54. By using Euclid's Division Lemma in 90 and 54, we get

$90=54\times 1+36$

Now, apply Euclid's Division Lemma in 54 and 36.

$54=36\times 1+18$

Now, apply Euclid's Division Lemma in 36 and 18.

$36=18\times 2+0$

Here, remainder is zero. So, the value of b in this step, i.e., 18, is the HCF of 54 and 90.

Therefore, the HCF of 54 and 90 is 18.