Question

LCM of two distinct natural numbers is 211. What is the HCF?

Answer 1

Hint: Here we check if the number as LCM is a prime number or a composite number. If it is a prime number then it has only two factors i.e. 1 and the number itself. Using the concept of HCF that HCF of two numbers is the factor of both numbers we find which of the numbers is HCF.
* LCM of two or more numbers is the least common multiple which is given by multiplication of prime factors of highest order.
* HCF of two or more numbers is the highest common factor which is given by the multiplication of all common prime numbers that are common in the prime factorization.

Complete step-by-step answer:
We are given that LCM of two numbers is 211.
211 is a prime number. So the only two factors of 211 are 1 and 211.
We first check if the LCM is a prime number. If the LCM is a prime number then we get the values of the two numbers.
We can write \[211 = 211 \times 1\]
Since, 211 has only two factors 1 and 211, so the number 211 is a prime number.
So the LCM of two numbers can be 211 if the two numbers have least common multiple as 211.
So, two numbers are 1 and 211.
Now we know HCF of two numbers is given by the highest common factor from the prime factorization of two numbers.
We write prime factorization of 1 and 211.
\[1 = 1 \times 1\]
\[211 = 1 \times 211\]
Common factor from the prime factorization of both numbers is 1.

HCF of two numbers is 1.

Note: Alternate Method:
We can solve the question using the formula for two numbers A and B,
\[A \times B = \]LCM of A, B \[ \times \]HCF of A, B
Here we are given LCM as 211
\[A \times B = 211 \times \]HCF of A, B
Dividing both sides by 211
\[\dfrac{{A \times B}}{{211}} = \] HCF of A, B
We write \[211 = 1 \times 211\] because that is the only way to factorize it
\[\dfrac{{A \times B}}{{211 \times 1}} = \] HCF of A, B
Since, we know HCF of two natural numbers is a natural number so LHS will be a natural number.
Therefore the numerator will be divisible by the denominator completely.
This can be possible if the numbers A and B are 211 and 1.
So, A is 211 and B is 1 or vice versa
HCF of 211 and 1 is the number which divides both the numbers.
So, 1 is the HCF of 211 and 1.