Question

Which of the following pairs are co-prime?
A.348 and 296
B.56 and 97
C.3025 and 4920
D.None

Answer 1

Hint: The common factor between the two numbers should be 1. C0-prime numbers are the set of numbers which do not have any common factor between them other than one that means their highest common factor (HCF) is 1.
In this question we will find the factors of the given pairs of numbers and then we will find their common factors and if their common factor is 1 then the pairs of numbers will be co-prime numbers otherwise not.

Complete step-by-step answer:
We will check all the given options one by one to find co prime
 A.348 and 296
Let us find the factor of \[348\]
 \[
   2\underline {\left| {348} \right.} \\
   2\underline {\left| {174} \right.} \\
   3\underline {\left| {87} \right.} \\
   29 \;
 \]
Hence \[\left( {348} \right) = 2 \times 2 \times 3 \times 29\]
Now the factor of \[296\]
 \[
   2\underline {\left| {296} \right.} \\
   2\underline {\left| {148} \right.} \\
   2\underline {\left| {74} \right.} \\
   37 \;
 \]
Hence \[\left( {296} \right) = 2 \times 2 \times 2 \times 37\]
So the Highest common factor of the numbers 348 and 296 will be
 \[HCF\left( {348,296} \right) = 2 \times 2\]
Hence we can say the pairs of numbers are not Coprime numbers.

B.56 and 97
Let us find the factor of \[56\]
 \[
   2\underline {\left| {56} \right.} \\
   2\underline {\left| {28} \right.} \\
   2\underline {\left| {14} \right.} \\
   7 \;
 \]
Hence \[\left( {56} \right) = 2 \times 2 \times 2 \times 7\]
Now the factor of \[97\] which is a prime number
 \[
   97\underline {\left| {97} \right.} \\
   1 \;
 \]
Hence \[\left( {97} \right) = 1\]
So the Highest common factor of the numbers 56 and 97 will be
 \[HCF\left( {56,97} \right) = 1\]
Hence we can say the pairs of numbers are Co-prime numbers since their common factor is 1.

C.3025 and 4920
Let us find the factor of \[3025\]
 \[
   5\underline {\left| {3025} \right.} \\
   5\underline {\left| {605} \right.} \\
   11\underline {\left| {121} \right.} \;
 \]
Hence \[\left( {3025} \right) = 5 \times 5 \times 11 \times 11\]
Now the factor of \[4920\] which is a prime number
 \[
   2\underline {\left| {4920} \right.} \\
   2\underline {\left| {2460} \right.} \\
   2\underline {\left| {1230} \right.} \\
   3\underline {\left| {615} \right.} \\
   5\underline {\left| {205} \right.} \\
   41 \;
 \]
Hence \[\left( {4920} \right) = 2 \times 2 \times 2 \times 3 \times 5 \times 41\]
So the Highest common factor of the numbers 3025 and 4920 will be
 \[HCF\left( {3025,4920} \right) = 5\]
Hence we can say the pairs of numbers are not Coprime numbers.
Therefore we can say option B is correct since the common factor of the number 56 and 97 is 1.
So, the correct answer is “Option B”.

Note: As we know since every prime number has only two factors which is 1 and the number itself so the only common factor of the two prime numbers will be 1 hence we can say the pair of two prime numbers will always be a co-prime number.
If we are given two numbers and it is said that the two numbers are co-prime numbers so we can directly say that the numbers are prime numbers.