Which of the following pairs of numbers are co-prime?

(a) 18 and 35

(b) 15 and 37

(c) 30 and 415

(d) 17 and 69

(e) 81 and 18

Question

Which of the following pairs of numbers are co-prime? (a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 69 (e) 81 and 18

Homi · Accepted Answer

Understand the Question

Two natural numbers are called co-prime if their only common factor is $1$ (that is, their $\text{HCF} = 1$ ).
To determine if a pair is co-prime:
- Find the factors of both numbers or test for common divisors.
- If the only common factor is $1$ , the pair is co-prime.
- If they share any common factor other than $1$ , they are not co-prime.

(a) 18 and 35

Step 1 · Find Common Factors of 18 and 35

Factors of $18 = 1, 2, 3, 6, 9, 18$

Factors of $35 = 1, 5, 7, 35$

The only common factor is $1$ .

Answer

(a) Co-prime

(b) 15 and 37

Step 1 · Find Common Factors of 15 and 37

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

Factors of $37 = 1, 37$ (since $37$ is a prime number)

The only common factor is $1$ .

Answer

(b) Co-prime

(c) 30 and 415

Step 1 · Check Common Factors of 30 and 415

Since $30$ ends in $0$ and $415$ ends in $5$ , both numbers are divisible by $5$ .

Because they share a common factor $5$ (which is other than $1$ ), they are not co-prime.

Answer

(c) Not co-prime

(d) 17 and 69

Step 1 · Find Common Factors of 17 and 69

Factors of $17 = 1, 17$ (since $17$ is a prime number)

Factors of $69 = 1, 3, 23, 69$

The only common factor is $1$ .

Answer

(d) Co-prime

(e) 81 and 18

Step 1 · Check Common Factors of 81 and 18

Both numbers are divisible by $3$ : $81 \div 3 = 27$ $18 \div 3 = 6$

Since they share a common factor other than $1$ , they are not co-prime.

Answer

(e) Not co-prime

Common Mistakes

Assuming numbers must be prime: Co-prime numbers do not have to be prime individually. For example, both $18$ and $35$ are composite, but they are co-prime because they share no common factors other than $1$ .
Overlooking divisibility shortcuts: Use quick divisibility tests (e.g., numbers ending in $0$ or $5$ share $5$ as a factor; digit sums divisible by $3$ ) instead of listing all factors.

Question 11