Question

Example of co-prime numbers from the below option is:
    (a) 25, 14
    (b) 18, 16
    (c) 9, 18
    (d) 11, 77

Answer 1

Hint: First of all, we need to know that co-prime numbers are the numbers which have only 1 as their common factor. The prime numbers which are multiplied together to give the original number are said to be the prime factors of the given number.

Complete step-by-step solution -
To solve this problem, we must find prime factors of each number separately.
Now, we are going to consider each pair given in the option and check whether they are co-prime numbers.
Let us consider the first pair 25, 14 given in option (a).
Now, on expressing 25 in terms of its prime factor, we get,
$25=5\times 5$
And, on expressing 14 in the same manner, we get,
$14=7\times 2$
There lies no common prime factor between 25 and 14. So, 1 is the only common factor between 25 and 14.
Hence, we can conclude that 25 and 14 are co-prime numbers.
Now, we will consider the second pair 18, 16 given in option (b).
Here, we are expressing 18 in terms of its prime factors,
$18=3\times 3\times 2$
And, on expressing 16 in the same way, we get,
$16=2\times 2\times 2\times 2$
Here, we can see that 2 is a common prime factor between 18 and 16. So, the second pair 18, 16 are not coprime numbers.
Let us consider the third pair 9, 18 given in option (c).
 Now, on expressing 9 in terms of its prime factors, we get,
$9=3\times 3$
And, on expressing 18 in the same way, we get,
$18=3\times 3\times 2$
Here, we can see that 3 is a common prime factor between 9 and 18. So, the third pair 9, 18 are not coprime numbers.
Now, let us consider the fourth pair 11, 77 given in option (d).
Since, 11 is a prime number, 11 is a prime factor of itself.
And, on expressing 77 in terms of its prime factors, we get,
$77=7\times 11$
Here, we can see that 11 is the common prime factor between 11 and 77. So, the fourth pair 11, 77 is not co-prime numbers.
Hence, the correct answer is option (a).
Note: We have found prime factors to check whether a given pair is co-prime. Instead, we can also list down the factors of given numbers in which 1 is one amidst it. Even factorization method leads us to right solution