Question

If a, b are co-prime, then ${{a}^{2}},{{b}^{2}}$ are?
a. Co-prime
b. Not coprime
c. Odd numbers
d. Even numbers

Answer 1

Hint: We will need to understand the concept of co-prime numbers to solve this question. We will take up some numeric values for the given co-prime numbers, a and b, like, a = 5 and b = 9 and then we will take their squares and check whether those numbers are co-prime or not, by taking the HCF of those numbers and if their HCF is 1, they are said to co-prime numbers, else not.

Complete step by step answer:
Here, in this question, we have been given that a and b are two numbers which are coprime and we need to check for ${{a}^{2}}\text{ and }{{b}^{2}}$.
Before we solve this question, we need to understand the concept of co-prime numbers. So, co-prime numbers are the two numbers which are positive integers and can be divided by only 1 factor, that is, 1.
Let us see how we can find whether two numbers are co-prime or not with the help of an example. We will follow a few steps in order to do so.
Step 1: We will find the HCF of the two numbers given.
Step 2: If their HCF is 1, we can say that they are co-prime numbers, else not.
So, let us consider two numbers as 5 and 9. So, the factors of 5 are 1 and 5, and that of 9 are 1, 3 and 9.
So, we can see that HCF of 5 and 9 is 1. Hence, we can say that they are co-prime numbers.
Now, in our question, it is given that a and b are two co-prime numbers and we have been asked to check whether ${{a}^{2}}\text{ and }{{b}^{2}}$ are co-prime are not.
So, we will take reference of the above example. So, let us consider a = 5 and b = 9.
So, we can write,
$\begin{align}
  & {{a}^{2}}={{5}^{2}}=5\times 5=25 \\
 & {{b}^{2}}={{9}^{2}}=9\times 9=81 \\
\end{align}$
Now, following the steps to check whether the numbers are co-prime or not, we will first find the HCF of 25 and 81.
So, the factors of 25 are 1, 5 and 25 and that of 81 are 1, 3, 9, 27 and 81.
So, here, we can see that the HCF of these two numbers is 1.
Hence, these two numbers are co-prime.
So, we can say that ${{a}^{2}}\text{ and }{{b}^{2}}$ are also co-prime.
Therefore, if a and b are co-prime, then ${{a}^{2}}\text{ and }{{b}^{2}}$ are also co-prime.

So, the correct answer is “Option A”.

Note: Here, instead of 5 and 9, the students can take any two numbers and follow the same steps to check whether they are co-prime or not. Also, the students must remember that co-prime numbers have no particular relation with the prime numbers. We can say that all two prime numbers are co-prime numbers but the vice – versa is not true.