Question

If the HCF of two numbers is 1, then the numbers are called:
A. Composite
B. Relatively prime or Co-prime
C. Perfect numbers
D. Irrational numbers

Answer 1

Hint:
1) HCF (Highest Common Factor) or GCD (Greatest Common Divisor) of a set of numbers is the greatest number which can divide all the numbers in the set.
2) 1 is a common factor of all the numbers.
3) If 1 is the "highest" common factor of two numbers, it means that no other number divides both of them, except 1.

Complete step by step solution:
Let us look at the definition / meaning of each number type given in the answer options:
A. Composite: Numbers which can be expressed as a product of two or more prime numbers. e.g. $20=2\times 2\times 5$. In contrast, prime numbers are the numbers which cannot be written as a product of two numbers except for 1 and itself. e.g. $19=1\times 19$.
B. Relatively prime or Co-prime: A pair of numbers are said to be relatively prime to each other (co-prime) if no number, except 1, can divide both of them. e.g. 8 and 15 are relatively prime, even though none of them is a prime number.
C. Perfect numbers: A Perfect number is a number whose sum of factors is equal to twice the number. e.g. 6 is a perfect number because the sum of its factors $1+2+3+6=12=2\times 6$ .
D. Irrational numbers: The numbers which cannot be expressed as a ratio of two integers are called irrational numbers. e.g. $\sqrt{2}=1.4142135...,\pi =3.141592..., e=2.71828...,\log 5=0.69897...$
From the above definitions, it is clear that the correct answer is B. Relatively prime or Co-prime.

Note:
1) A pair of prime numbers is always coprime.
2) For a set of more than two numbers, the term used is "pairwise coprime", which means that no number can divide all of them.
3) 1 is a factor of every number and 0 is a multiple of every number.
4) Only 51 perfect numbers are known till date, even though it is believed (not proved) that there are infinitely many of them.
5) Irrational numbers are non-terminating and non-repeating when expressed in decimal form. Rational numbers on the other hand either terminate or are non-terminating and repeating when expressed in decimal form. e.g. $\dfrac{3}{5}=0.6,\dfrac{4}{11}=0.363636...=0.\overline{36}$