Find the HCF of the following numbers: (a) 24, 180 (b) 42, 75, 24 (c) 240, 378 (d) 400, 2500 (e) 300, 800

Understand the Question The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides each of them without leaving a remainder. To find the HCF using prime factorisation: 1. Express each number as a product of its prime factors. 2. Identify the prime factors common to all the given numbers. 3. Multiply the lowest powers of these common prime factors together. Part (a) 24, 180 Step 1 · Find HCF of 24 and 180 Prime factors of : Prime factors of : Common prime factors are , , and . Answer > (a) Part (b) 42, 75, 24 Step 1 · Find HCF of 42, 75, and 24 Prime factors of : Prime factors of : Prime factors of : The only common prime factor in all three numbers is . Answer > (b) Part (c) 240, 378 Step 1 · Find HCF of 240 and 378 Prime factors of : Prime factors of : Common prime factors are and . Answer > (c) Part (d) 400, 2500 Step 1 · Find HCF of 400 and 2500 Prime factors of : Prime factors of : Common prime factors are , , , and . Answer > (d) Part (e) 300, 800 Step 1 · Find HCF of 300 and 800 Prime factors of : Prime factors of : Common prime factors are , , , and . Answer > (e) Common Mistakes Partial Common Factors in 3 Numbers: In part (b), taking as a common factor because it divides and , but forgetting that it does not divide . For HCF, a prime factor must divide all numbers in the group. Confusing HCF with LCM: Taking the highest powers of prime factors instead of the lowest common powers.

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Question 4

Find the HCF of the following numbers:

(a) 24, 180

(b) 42, 75, 24

(d) 400, 2500

(e) 300, 800

Solution

Understand the Question

The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides each of them without leaving a remainder.
To find the HCF using prime factorisation:
1. Express each number as a product of its prime factors.
2. Identify the prime factors common to all the given numbers.
3. Multiply the lowest powers of these common prime factors together.

(a) 24, 180

Step 1 · Find HCF of 24 and 180

Diagram 1

Prime factors of $24$ :

\begin{aligned} 24 &= 2 \times 12 \\ &= 2 \times 2 \times 6 \\ &= 2 \times 2 \times 2 \times 3 \end{aligned}

Prime factors of $180$ :

\begin{aligned} 180 &= 2 \times 90 \\ &= 2 \times 2 \times 45 \\ &= 2 \times 2 \times 3 \times 15 \\ &= 2 \times 2 \times 3 \times 3 \times 5 \end{aligned}

Common prime factors are $2$ , $2$ , and $3$ .

\begin{aligned} \text{HCF}(24, 180) &= 2 \times 2 \times 3 \\ &= 12 \end{aligned}

Answer

(a) $12$

(b) 42, 75, 24

Step 1 · Find HCF of 42, 75, and 24

Diagram 2

Prime factors of $42$ :

\begin{aligned} 42 &= 2 \times 21 \\ &= 2 \times 3 \times 7 \end{aligned}

Prime factors of $75$ :

\begin{aligned} 75 &= 3 \times 25 \\ &= 3 \times 5 \times 5 \end{aligned}

Prime factors of $24$ :

\begin{aligned} 24 &= 2 \times 12 \\ &= 2 \times 2 \times 6 \\ &= 2 \times 2 \times 2 \times 3 \end{aligned}

The only common prime factor in all three numbers is $3$ . $\text{HCF}(42, 75, 24) = 3$

Answer

(b) $3$

(c) 240, 378

Step 1 · Find HCF of 240 and 378

Diagram 3

Prime factors of $240$ :

\begin{aligned} 240 &= 2 \times 120 \\ &= 2 \times 2 \times 60 \\ &= 2 \times 2 \times 2 \times 30 \\ &= 2 \times 2 \times 2 \times 2 \times 15 \\ &= 2 \times 2 \times 2 \times 2 \times 3 \times 5 \end{aligned}

Prime factors of $378$ :

\begin{aligned} 378 &= 2 \times 189 \\ &= 2 \times 3 \times 63 \\ &= 2 \times 3 \times 3 \times 21 \\ &= 2 \times 3 \times 3 \times 3 \times 7 \end{aligned}

Common prime factors are $2$ and $3$ .

\begin{aligned} \text{HCF}(240, 378) &= 2 \times 3 \\ &= 6 \end{aligned}

Answer

(c) $6$

(d) 400, 2500

Step 1 · Find HCF of 400 and 2500

Diagram 4

Prime factors of $400$ :

\begin{aligned} 400 &= 2 \times 200 \\ &= 2 \times 2 \times 100 \\ &= 2 \times 2 \times 2 \times 50 \\ &= 2 \times 2 \times 2 \times 2 \times 25 \\ &= 2 \times 2 \times 2 \times 2 \times 5 \times 5 \end{aligned}

Prime factors of $2500$ :

\begin{aligned} 2500 &= 2 \times 1250 \\ &= 2 \times 2 \times 625 \\ &= 2 \times 2 \times 5 \times 125 \\ &= 2 \times 2 \times 5 \times 5 \times 25 \\ &= 2 \times 2 \times 5 \times 5 \times 5 \times 5 \end{aligned}

Common prime factors are $2$ , $2$ , $5$ , and $5$ .

\begin{aligned} \text{HCF}(400, 2500) &= 2 \times 2 \times 5 \times 5 \\ &= 4 \times 25 \\ &= 100 \end{aligned}

Answer

(d) $100$

(e) 300, 800

Step 1 · Find HCF of 300 and 800

Diagram 5

Prime factors of $300$ :

\begin{aligned} 300 &= 2 \times 150 \\ &= 2 \times 2 \times 75 \\ &= 2 \times 2 \times 3 \times 25 \\ &= 2 \times 2 \times 3 \times 5 \times 5 \end{aligned}

Prime factors of $800$ :

\begin{aligned} 800 &= 2 \times 400 \\ &= 2 \times 2 \times 200 \\ &= 2 \times 2 \times 2 \times 100 \\ &= 2 \times 2 \times 2 \times 2 \times 50 \\ &= 2 \times 2 \times 2 \times 2 \times 2 \times 25 \\ &= 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \end{aligned}

Common prime factors are $2$ , $2$ , $5$ , and $5$ .

\begin{aligned} \text{HCF}(300, 800) &= 2 \times 2 \times 5 \times 5 \\ &= 4 \times 25 \\ &= 100 \end{aligned}

Answer

(e) $100$

Common Mistakes

Partial Common Factors in 3 Numbers: In part (b), taking $2$ as a common factor because it divides $42$ and $24$ , but forgetting that it does not divide $75$ . For HCF, a prime factor must divide all numbers in the group.
Confusing HCF with LCM: Taking the highest powers of prime factors instead of the lowest common powers.

Question 4

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