Question 5
Q5: If A and B are positive integers greater than 1 such that the GCD of A and B is 1, then which of the following must be true?
I. A and B are prime numbers.
II. A and B are consecutive numbers.
III. A and B do not have a common prime factor
IV. The product AB has two prime factors
V. A and B have the opposite even-odd nature
- I & II only
- III only
- I, V only
- I, II, III, V only
- I, II, III, IV & V
Correct Answer
B
Solution
Given
We are given two positive integers A & B which are greater than 1. We are also given that GCD(A, B) = 1. We are asked to find the options which must be true
Approach
We know that if two positive integers have their GCD as 1, they do not have any common prime factor. We will use this understanding to find out the must be true statements out of the statements given
Working Out
Let’s evaluate the statements given in the option:
I. A & B are prime numbers – May not be true.GCD of two numbers is 1 when they don’t have any common prime factor. These numbers need not necessarily be primes. For example 4, 9 have their GCD = 1 but they are not prime numbers.
II. A and B are consecutive numbers– May not be true. Two numbers having their GCD = 1 need not necessarily be consecutive numbers.
III. A & B do not have a common prime factor– Must be true. Numbers who have their GCD = 1 do not have any common prime factor.
IV. The product AB has two prime factors– May not be true. Number A and number B can have more than 1 different prime numbers as their factors. Thus, the product of A & B may have more than 2 prime factors
V. A & B have the opposite even-odd nature– May not be true. Let us consider A = 7 and B = 9. In this case, both A and B are odd numbers greater than 1 and their GCD is 1. Hence, it is not necessarily true that A & B have the opposite even-odd nature.
Answer: Option (B)
