Question

What is the H.C.F. of two co-prime numbers?
(A) $1$
(B) $0$
(C) $2$
(D) None of these

Answer 1

Hint: The full form of H.C.F. is the highest common factor. It is defined as the largest possible number which exactly divides the given pair of numbers. The highest common factor is calculated by multiplying all the factors which appear in both lists. For example the highest common factor of 24 and 36 is 12 because 12 is the highest common factor of 24 and 36. Now let us discuss the given problem.

Complete step by step answer:
Let us assume p and q to be any two prime numbers, taken in general, where $p \ne q$ .
Then for the prime number p, the only factors of p are 1 and p itself.
Again for the prime number q, the only factors of it are 1 and q itself. Now we find the common factors of p and q. We can see, the only common factor that exists in this case is 1, as $p \ne q$ and p and q are relatively prime. Then the only common factor we get is 1.
Therefore the H.C.F. of the two prime p and q will be 1 as 1 is the only and highest common divisor or factor of the two prime p and q [because $p \ne q$ ].
Now since p and q are chosen arbitrarily, then we have, for any two co-prime numbers, there H.C.F. will be 1.

So, the correct answer is “Option A”.

Note:
Let us have a brief concept about the prime numbers. A prime number is defined as the numbers that are divisible only by 1 and itself. So if we take a number “a” as a prime number, then the only divisors of “a” are 1 and a itself. Note that, in Mathematics “1” is not treated as a prime number. Some of the most common prime numbers are 2,3,5,7,11,13,17,19,23,29,… etc.