Question

Which of the following numbers are co-prime:

• A. $18\mathrm{and}35$
• B. $15\mathrm{and}37$
• C. $30\mathrm{and}415$
• D. $17\mathrm{and}168$
• E. $216\mathrm{and}215$
• F. $81\mathrm{and}16$

Answer 1

Answer: (A), (B), (E), (F)

$81\mathrm{and}16$

Step 1. Explanation for the correct options.

$18\mathrm{and}35$.

• Factors of $18=1,2,3,6,9,18$
• Factors of $35=1,5,7,35$

Common factor of $18\mathrm{and}35$ is $1$.

Since both have only one common factor, i.e., $1$, therefore, they are co-prime numbers.

$15\mathrm{and}37$

• Factors of $15=1,3,5,15$
• Factors of $37=1,37$

Common factor of $15\mathrm{and}37$ is $1$.

Since both have only one common factor, i.e., $1$, therefore, they are co-prime numbers.

$216\mathrm{and}215$

• Factors of $216=1,2,3,4,6,8,36,72,108,216$
• Factors of $215=1,5,43,215$

Common factor of $216\mathrm{and}215$ is $1$.

Since both have only one common factor, i.e., $1$, therefore, they are co-prime numbers.

$81\mathrm{and}16$

• Factors of $81=1,3,9,27,81$
• Factors of $16=1,2,4,8,16$

Common factor of $81\mathrm{and}16$ is $1$.

Since both have only one common factor, i.e., $1$, therefore, they are co-prime numbers.

Step 2. Explanation for the incorrect options.

$30\mathrm{and}415$

Factors of $30=1,2,3,5,6,15,30$

Factors of $415=1,5,83,415$

Common factor of $30\mathrm{and}415=1,5$

Since both have more than one common factor, therefore, they are not co-prime numbers.

$17\mathrm{and}168$

• Factors of $17=1,17$
• Factors of $68=1,2,4,17,34,86$

Common factor of $17\mathrm{and}168=1,17$

Since both have more than one common factor. Therefore, they are not co-prime numbers.

Hence, the correct options are $\mathrm{A},\mathrm{B},\mathrm{E}\mathrm{and}\mathrm{F}$.