Question

HCF of 26 and 91 is:
(a) 13
(b) 2366
(c) 91
(d) 182

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorisation of a number is representing the number as the product of prime numbers. We write the prime factorisation for all the given numbers then we use the theorem that the HCF of two numbers in a prime factorisation is given by taking the common primes from the prime factorisation.
That is the two numbers are in the form
\[x={{2}^{a}}\times {{3}^{b}}\times {{7}^{c}}\]
\[y={{2}^{k}}\times {{5}^{l}}\times {{7}^{m}}\]
Here the HCF of \[x,y\] is given as
\[\Rightarrow HCF={{2}^{\left| a-k \right|}}\times {{7}^{\left| c-m \right|}}\]
Here we have modulus because we don’t know that which is greater.

Complete step-by-step answer:
We are asked to find the HCF of 26 and 91
Let us use the prime factorisation for the number 26.
We know that the prime factorisation of a number is representing the number as the product of prime numbers.
So, let us represent the number 26 in terms of first prime number 2 then we get
\[\Rightarrow 26=2\times 13\]
Here, we can see that the number 13 is a prime number.
So, we can say that the prime factorisation of number 26 is complete.
Now, let us use the prime factorisation for the second number that is 91
Here, we can see that the number 91 cannot be represented in terms of 2 because it is an odd number.
Now, let us represent the number 91 by using the second prime number that is 3 then we get
\[\Rightarrow 91=7\times 13\]
Here, we can see that the number 13 is a prime number.
So, we can say that the prime factorisation of number 91 is complete.
Let us assume that the HCF of 26 and 91 as \['N'\]
We know that the HCF of two numbers in a prime factorisation is given by taking the common primes from the prime factorisation.
Now, let us take the common primes from the prime factorisation of 26 and 91 then we get
\[\Rightarrow N=13\]
Therefore, we can conclude that the HCF of 26 and 91 is 13.

So, the correct answer is “Option A”.

Note: We have another method for finding the HCF that is the division method.
In the division method we divide the large number with the small number to obtain the remainder. Then again we divide the divisor used in the above division with the remainder to get the next remainder. We carry on this process until we get remainder as 0. Then the divisor used when we got the remainder as 0 as the HCF of given numbers.
Now, let us divide the number 91 with 26 then we get
\[\Rightarrow \dfrac{91}{26}=3+\dfrac{13}{26}\]
Here we can see that the remainder is 13 and the divisor is 26
Now, by dividing the number 26 with 13 we get
\[\Rightarrow \dfrac{26}{13}=2+\dfrac{0}{13}\]
Here, we can see that the remainder is 0 when the divisor is 13
So, we can conclude that the HCF of 26 and 91 is 13.