Question

Use Euclids division algorithm to find the HCF of 441, 567, 693.

Answer 1

Hint: We are given three numbers and we are asked to find the HCF of these number using the Euclids division algorithm. Euclidean algorithm states that for finding HCF of numbers A and B is given by – if A = 0, then the HCF of the nos. is B and vice versa. Writing A in quotient form, we have A = BQ + R. Using this quotient form, we will find the HCF taking two numbers at a time, that is, \[567=441\times 1+126\] and we continue till we get zero. Then, we will take the HCF of the quotient from the previous one and the number 693 and hence, we will have the HCF of the three given numbers.

Complete step-by-step answer:
According to the given question, we are given three numbers 441, 567, 693 and we have to find the HCF of these numbers. And to find the HCF we are to use the Euclids division algorithm.
As per Euclids division algorithm, to find the HCF of two numbers say, A and B, we have,
If A = 0, then HCF of A and B is B. Similarly, if B = 0, then the HCF of A and B is A.
If we are to write the quotient form for A, we get,
A = BQ + R
We will be using this equation in order to find the HCF of the given numbers.
First, we will find the HCF of the numbers 441 and 567. Writing the number 567 using the quotient form, we have,
\[567=441\times 1+126\]
\[\Rightarrow 126=63\times 2+0\]
Since, we now have the remainder as 0,
So, the HCF of 441 and 567 is \[HCF(441,567)=63\].
Next, we will find the HCF of 63 and 693.
\[693=63\times 11+0\]
Since, the remainder is 0
So, the \[HCF(63,693)=63\].
Therefore, the HCF of the given three number 441, 567 and 693 is,
 \[HCF(441,567,693)=63\]

Note: The Euclidean algorithm is to be used explicitly as it is asked in the question, so do not use the usual factorization to find the HCF. Also, the quotient form of numbers under consideration should carefully written without any errors.