Question

Which of these are not coprime numbers?
A) $32,24$
B) $25,32$
C) $15,22$
D) $18,35$

Answer 1

Hint: Let us first know what co-prime numbers are. Coprime numbers are those numbers, who do not have any factor, other than $1$, common between them. In other words, two numbers are coprime if their Greatest Common Factor or GCF is $1$. So, to solve this problem we will use the method of optional analysis. We will find the prime factors of every number in each option and if there are any common prime factors between them, then they are not co-primes, but if they don’t have any common prime factors between them, then they are co-primes. So, let us see how to solve this problem.

Complete step by step answer:
Let’s analyze the given options.

Option (1):
Factors of $32 = 2 \times 2 \times 2 \times 2 \times 2$.
And, factors of $24 = 2 \times 2 \times 2 \times 3$
Therefore, there are common factors between $32$ and $24$, that is, $2 \times 2 \times 2 = 8$.
Hence, $32$ and $24$ are not co-prime numbers.

Option (B):
Factors of $25 = 5 \times 5$.
And, factors of $32 = 2 \times 2 \times 2 \times 2 \times 2$
Therefore, there are no common factors between $25$ and $32$.
Hence, $25$ and $32$ are co-prime numbers.

Option (C):
Factors of $15 = 3 \times 5$.
And, factors of $22 = 2 \times 11$
Therefore, there are no common factors between $15$ and $22$.
Hence, $15$ and $22$ are co-prime numbers.

Option (D):
Factors of $18 = 2 \times 3 \times 3$.
And, factors of $35 = 5 \times 7$
Therefore, there are no common factors between $18$ and $35$.
Hence, $18$ and $35$ are co-prime numbers.

Therefore, $32$ and $24$ are not co-prime numbers. So, option (A) is correct.

Note:
We can easily conclude whether some numbers are co-primes or not. If both the numbers are even numbers, then, they are not co-primes, as they will surely have $2$ as a factor to both of them. If both the numbers have $5$ at their one’s place, then both of them have $5$ as a factor, hence they can’t be co-primes. And, also, if two numbers have $0$ in their one’s place, then both of them have $2$ and $5$ as their factors, and hence will not be co-primes.