Finding Common Ground | FIO

Question 2

Find the common factors and the HCF of the following numbers:

(a) 50, 60

(b) 140, 275

(d) 370, 592

(e) 81, 243

Solution

Understand the Question

To find the common factors and the Highest Common Factor (HCF) of two numbers:

Prime Factorisation: Express each number as a product of its prime factors.
List All Factors: Form all possible factors by taking combinations of the prime factors (along with $1$ and the number itself).
Common Factors: Identify the numbers present in both lists of factors.
Highest Common Factor (HCF): The largest number among the common factors (which also equals the product of the common prime factors).

(a) 50, 60

Step 1 · Prime factorise 50 and 60

Prime factorisation of $50$ : $50 \div 2 = 25$ $25 \div 5 = 5$ $5 \div 5 = 1$ $50 = 2 \times 5 \times 5$

Prime factorisation of $60$ : $60 \div 2 = 30$ $30 \div 2 = 15$ $15 \div 3 = 5$ $5 \div 5 = 1$ $60 = 2 \times 2 \times 3 \times 5$

Step 2 · Find Common Factors and HCF

List all factors of each number:

Factors of $50$ : $1, 2, 5, 10, 25, 50$
Factors of $60$ : $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

Common factors are the numbers present in both lists: $\text{Common factors} = 1, 2, 5, 10$

Calculate HCF from common prime factors ( $2$ and $5$ ): $\text{HCF}(50, 60) = 2 \times 5 = 10$

Answer

(a) Common factors: $1, 2, 5, 10$ ; $\text{HCF} = 10$

(b) 140, 275

Step 1 · Prime factorise 140 and 275

Prime factorisation of $140$ : $140 \div 2 = 70$ $70 \div 2 = 35$ $35 \div 5 = 7$ $7 \div 7 = 1$ $140 = 2 \times 2 \times 5 \times 7$

Prime factorisation of $275$ : $275 \div 5 = 55$ $55 \div 5 = 11$ $11 \div 11 = 1$ $275 = 5 \times 5 \times 11$

Step 2 · Find Common Factors and HCF

List all factors of each number:

Factors of $140$ : $1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$
Factors of $275$ : $1, 5, 11, 25, 55, 275$

Common factors: $\text{Common factors} = 1, 5$

Calculate HCF from the common prime factor: $\text{HCF}(140, 275) = 5$

Answer

(b) Common factors: $1, 5$ ; $\text{HCF} = 5$

(c) 77, 725

Step 1 · Prime factorise 77 and 725

Prime factorisation of $77$ : $77 \div 7 = 11$ $11 \div 11 = 1$ $77 = 7 \times 11$

Prime factorisation of $725$ : $725 \div 5 = 145$ $145 \div 5 = 29$ $29 \div 29 = 1$ $725 = 5 \times 5 \times 29$

Step 2 · Find Common Factors and HCF

List all factors of each number:

Factors of $77$ : $1, 7, 11, 77$
Factors of $725$ : $1, 5, 25, 29, 145, 725$

Common factors: $\text{Common factor} = 1$

Since there are no common prime factors, the HCF is $1$ : $\text{HCF}(77, 725) = 1$

Answer

(c) Common factor: $1$ ; $\text{HCF} = 1$

(d) 370, 592

Step 1 · Prime factorise 370 and 592

Prime factorisation of $370$ : $370 \div 2 = 185$ $185 \div 5 = 37$ $37 \div 37 = 1$ $370 = 2 \times 5 \times 37$

Prime factorisation of $592$ : $592 \div 2 = 296$ $296 \div 2 = 148$ $148 \div 2 = 74$ $74 \div 2 = 37$ $37 \div 37 = 1$ $592 = 2 \times 2 \times 2 \times 2 \times 37$

Step 2 · Find Common Factors and HCF

List all factors of each number:

Factors of $370$ : $1, 2, 5, 10, 37, 74, 185, 370$
Factors of $592$ : $1, 2, 4, 8, 16, 37, 74, 148, 296, 592$

Common factors: $\text{Common factors} = 1, 2, 37, 74$

Calculate HCF from common prime factors ( $2$ and $37$ ): $\text{HCF}(370, 592) = 2 \times 37 = 74$

Answer

(d) Common factors: $1, 2, 37, 74$ ; $\text{HCF} = 74$

(e) 81, 243

Step 1 · Prime factorise 81 and 243

Prime factorisation of $81$ : $81 \div 3 = 27$ $27 \div 3 = 9$ $9 \div 3 = 3$ $3 \div 3 = 1$ $81 = 3 \times 3 \times 3 \times 3$

Prime factorisation of $243$ : $243 \div 3 = 81$ $81 \div 3 = 27$ $27 \div 3 = 9$ $9 \div 3 = 3$ $3 \div 3 = 1$ $243 = 3 \times 3 \times 3 \times 3 \times 3$

Step 2 · Find Common Factors and HCF

List all factors of each number:

Factors of $81$ : $1, 3, 9, 27, 81$
Factors of $243$ : $1, 3, 9, 27, 81, 243$

Common factors: $\text{Common factors} = 1, 3, 9, 27, 81$

Calculate HCF from common prime factors ( $3 \times 3 \times 3 \times 3$ ): $\text{HCF}(81, 243) = 3 \times 3 \times 3 \times 3 = 81$

Answer

(e) Common factors: $1, 3, 9, 27, 81$ ; $\text{HCF} = 81$

Common Mistakes

Stating HCF as 0 for Coprime Numbers: When two numbers have no common prime factors (such as $77$ and $725$ ), their common factor and HCF is always $1$ , never $0$ .
Missing 1 as a Factor: Forgetting that $1$ is a factor of every number and is always listed among the common factors.
Missing Composite Factors: Forgetting to combine prime factors when listing all factors of a number (e.g. for $50 = 2 \times 5 \times 5$ , missing $2 \times 5 = 10$ or $5 \times 5 = 25$ ).

Question 2

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