The first number has prime factorisation and the second number has prime factorisation . Are they co-prime? Does one of them divide the other?

Understand the Question Co-prime numbers: Two numbers are co-prime if their only common factor is (i.e. they share no common prime factors). Divisibility via prime factors: A number divides a number if and only if all prime factors of (including their powers) are present in the prime factorisation of . We are given: Step 1 · Check if the Numbers are Co-prime Given prime factorisations: Common prime factors are and . Since the numbers have common factors other than , they are not co-prime. Step 2 · Check Divisibility Between the Numbers For one number to divide another, all of its prime factors must be present in the other number. Does the first number divide the second number? The first number contains the prime factor , which is not in the second number (). Therefore, the first number does not divide the second number. Does the second number divide the first number? The second number contains the prime factor , which is not in the first number (). Therefore, the second number does not divide the first number. Hence, neither number divides the other. Answer > 1. No, they are not co-prime (they share common prime factors and ). 2. No, neither number divides the other. Common Mistakes Misunderstanding Co-prime Numbers: Assuming that the numbers themselves must be prime. Co-prime simply means their . Confusing Common Factors with Divisibility: Assuming that because they share prime factors ( and ), one must divide the other. A number divides another only if all its prime factors are contained in the dividend.

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Question 34

The first number has prime factorisation $2 \times 3 \times 7$ and the second number has prime factorisation $3 \times 7 \times 11$ . Are they co-prime? Does one of them divide the other?

Solution

Understand the Question

Co-prime numbers: Two numbers are co-prime if their only common factor is $1$ (i.e. they share no common prime factors).
Divisibility via prime factors: A number $A$ divides a number $B$ if and only if all prime factors of $A$ (including their powers) are present in the prime factorisation of $B$ .
We are given:
- $\text{First number} = 2 \times 3 \times 7$
- $\text{Second number} = 3 \times 7 \times 11$

Step 1 · Check if the Numbers are Co-prime

Given prime factorisations: $\text{First number} = 2 \times 3 \times 7$ $\text{Second number} = 3 \times 7 \times 11$

Common prime factors are $3$ and $7$ . $\text{HCF} = 3 \times 7 = 21 \neq 1$

Since the numbers have common factors other than $1$ , they are not co-prime.

Step 2 · Check Divisibility Between the Numbers

For one number to divide another, all of its prime factors must be present in the other number.

Does the first number divide the second number? The first number contains the prime factor $2$ , which is not in the second number ( $3 \times 7 \times 11$ ). Therefore, the first number does not divide the second number.
Does the second number divide the first number? The second number contains the prime factor $11$ , which is not in the first number ( $2 \times 3 \times 7$ ). Therefore, the second number does not divide the first number.

Hence, neither number divides the other.

Answer

No, they are not co-prime (they share common prime factors $3$ and $7$ ).

No, neither number divides the other.

Common Mistakes

Misunderstanding Co-prime Numbers: Assuming that the numbers themselves must be prime. Co-prime simply means their $\text{HCF} = 1$ .
Confusing Common Factors with Divisibility: Assuming that because they share prime factors ( $3$ and $7$ ), one must divide the other. A number divides another only if all its prime factors are contained in the dividend.

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