Question

The co – primes from the following pairs are _____
(a) 7 and 63
(b) 36 and 25
(c) 35 and 21
(d) 63 and 81

Answer 1

Hint: We solve this problem by using the prime factorisation method for each pair in the given options.
The prime factorisation method is the method of representing the given number as the product of prime numbers.
Then we check whether the numbers are co – primes or not. The co – primes are the pair of numbers that have only one common factor and that is 1.

Complete step by step answer:
We are asked to find the pair which are co – primes from the given pairs.
Let us check each and every option one by one.
(a) 7 and 63
Now, let us use the prime factorisation for the number 7
Here, we can see that the number 7 is a prime number
So, we can take the prime factorisation for 7 as
\[\Rightarrow 7=1\times 7\]
Now, let us use the prime factorisation method for 63 then we get
\[\Rightarrow 63={{3}^{2}}\times 7\]
Here, we can see that there is common prime 7 in the prime factorisation of 7 and 63
We know that the co – primes are the pair of numbers having only one common factor that is 1
By using the above condition we can conclude that the pair 7 and 63 are not co – primes.
(b) 36 and 25
Now, let us use the prime factorisation method for 36 then we get
\[\Rightarrow 36={{2}^{2}}\times {{3}^{2}}\]
Now, let us use the prime factorisation method for 25 then we get
\[\Rightarrow 25={{5}^{2}}\]
Here, we can see that there are no common factors for 36 and 25
By using the co – primes definition we can conclude that the pair 36 and 25 is co – primes.
(c) 35 and 21
Now, let us use the prime factorisation method for 35 then we get
\[\Rightarrow 35=5\times 7\]
Now, let us use the prime factorisation method for 21 then we get
\[\Rightarrow 21=3\times 7\]
Here, we can see that there is common factor for 35 and 21 that is 7
By using the co – primes definition we can conclude that the pair 35 and 21 is not co – primes.
(d) 63 and 81
Now, let us use the prime factorization method for 63 then we get
\[\Rightarrow 63={{3}^{2}}\times 7\]
Now, let us use the prime factorization method for 81 then we get
\[\Rightarrow 81={{3}^{4}}\]
Here, we can see that there are no common factors for 63 and 81 that is 3
By using the co – primes definition we can conclude that the pair 63 and 81 is not co – primes.
Therefore, option (b) is the correct answer.

Note:
Students may make mistakes in understanding the definition of co – primes.
We know that co – primes have only one common factor.
This common factor should be only 1
But students may do mistake and take the common factor is other than 1
This gives the wrong answer because co – primes have only 1 as the common factor.