Question

# Find the HCF of 336 and 54 using Euclid’s division lemma.

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Hint: Euclid’s division lemma can be used to calculate HCF of big numbers easily, comparative to other methods. As in this method the denominator is used to take the form of dividend while splitting the divisor.

Here, we have to calculate HCF of 336 and 54 using Euclid’s division lemma method, where
Euclid’s division lemma states that, if two positive integers such as ‘a’ and ‘b’, then there exists unique integers ‘q’ and ‘r’ which satisfies the condition as $a=bq+r$, where $0\le r\le b$.
Main aim of Euclid’s division lemma is to calculate HCF of two positive integers, where HCF is the largest number which exactly divides two or more positive integers.
Here, denominator is used as a dividend to split the numerator, along with the remainder.
This process is continued till the remainder comes out be 0. As and when the remainder becomes 0, the dividend of that equation becomes HCF of the given two numbers.
Now, applying same concept with given set of equations, we have
Since, $336>54$, we can apply the Euclid’s division lemma, i.e.,
$336=54\times 6+12$
Now, as remainder came out to be 12, which is not equal to 0, then again, we have
$54=12\times 4+6$
Similarly, $6\ne 0$
Again, $12=6\times 2+0$
As in above equation remainder is equal to 0, then we can say that 6 is the required divisor which can completely divide both the dividends 336 and 54, with remainder equal to 0.
Hence, we can say that HCF of 336 and 54 is 6, using the Euclid’s division lemma method.

Note: This type of HCF finding problems can very easily be solved using the long-division method or prime factorisation method. Those methods can be used to cross-check the answers of Euclid’ division lemma.