Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.

a. 30 and 45

b. 57 and 85

c. 121 and 1331

d. 343 and 216

Question

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45 b. 57 and 85 c. 121 and 1331 d. 343 and 216

Homi · Accepted Answer

Understand the Question

Two numbers are co-prime (or relatively prime) if their only common factor is $1$ , meaning they do not share any common prime factors.
Approach:
1. Guess using divisibility tests (e.g., checking if both numbers are even, divisible by $3$ , or end in $0$ / $5$ ).
2. Verify by finding the prime factorisation of each number and checking for shared prime factors.

a. 30 and 45

Step 1 · Prime Factorisation and Verification

Both numbers end in $0$ or $5$ , so they are divisible by $5$ . The sum of digits of $30$ is $3$ , and for $45$ is $9$ , so both are also divisible by $3$ .

Guess: Not co-prime.

Prime factorisation of $30$ :

\begin{aligned} 30 &= 2 \times 15 \\ &= 2 \times 3 \times 5 \end{aligned}

Prime factorisation of $45$ :

\begin{aligned} 45 &= 3 \times 15 \\ &= 3 \times 3 \times 5 \end{aligned}

Common prime factors $= 3, 5$ .

Since they share common prime factors other than $1$ , $30$ and $45$ are not co-prime.

Answer

a. Not co-prime

b. 57 and 85

Step 1 · Prime Factorisation and Verification

The sum of digits of $57$ is $5 + 7 = 12$ (divisible by $3$ ). $85$ ends in $5$ (divisible by $5$ ). Their prime factors appear distinct.

Guess: Co-prime.

Prime factorisation of $57$ : $57 = 3 \times 19$

Prime factorisation of $85$ : $85 = 5 \times 17$

Common prime factors $= \text{None}$

Since they have no common prime factors other than $1$ , $57$ and $85$ are co-prime.

Answer

b. Co-prime

c. 121 and 1331

Step 1 · Prime Factorisation and Verification

We know $121 = 11 \times 11$ and $1331 \div 11 = 121$ .

Guess: Not co-prime.

Prime factorisation of $121$ : $121 = 11 \times 11$

Prime factorisation of $1331$ :

\begin{aligned} 1331 &= 11 \times 121 \\ &= 11 \times 11 \times 11 \end{aligned}

Common prime factor $= 11$ .

Since they share the common prime factor $11$ , $121$ and $1331$ are not co-prime.

Answer

c. Not co-prime

d. 343 and 216

Step 1 · Prime Factorisation and Verification

$343$ is odd (not divisible by $2$ , $3$ , or $5$ , but divisible by $7$ ). $216$ is an even number divisible by $2$ and $3$ .

Guess: Co-prime.

Prime factorisation of $343$ :

\begin{aligned} 343 &= 7 \times 49 \\ &= 7 \times 7 \times 7 \end{aligned}

Prime factorisation of $216$ :

\begin{aligned} 216 &= 2 \times 108 \\ &= 2 \times 2 \times 54 \\ &= 2 \times 2 \times 2 \times 27 \\ &= 2 \times 2 \times 2 \times 3 \times 9 \\ &= 2 \times 2 \times 2 \times 3 \times 3 \times 3 \end{aligned}

Common prime factors $= \text{None}$

Since they share no common prime factors other than $1$ , $343$ and $216$ are co-prime.

Answer

d. Co-prime

Common Mistakes

Co-prime vs. Prime Numbers: Thinking that individual numbers in a pair must be prime to be co-prime. Composite numbers (like $343$ and $216$ ) can also be co-prime to each other if they share no common factor other than $1$ .
Missing Shared Factors: Stopping factorisation too early and missing a common larger prime factor (such as $11$ in part c).

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