Find the LCM of the following numbers: (a) 30, 72 (b) 36, 54 (c) 105, 195, 65 (d) 222, 370

Understand the Question To find the Least Common Multiple (LCM) of a set of numbers using the prime factorisation method: 1. Find the prime factors of each number and express them in exponential form. 2. Identify all unique prime factors across the numbers. 3. Take the highest power of each prime factor and multiply them together. Part (a) 30, 72 Step 1 · Prime Factorisation of 30 and 72 For : For : Step 2 · Calculate LCM Taking the highest power of each prime factor (, , ): Answer > (a) 360 Part (b) 36, 54 Step 1 · Prime Factorisation of 36 and 54 For : For : Step 2 · Calculate LCM Taking the highest power of each prime factor (, ): Answer > (b) 108 Part (c) 105, 195, 65 Step 1 · Prime Factorisation of 105, 195, and 65 For : For : For : Step 2 · Calculate LCM Taking the highest power of each prime factor (, , , ): Answer > (c) 1365 Part (d) 222, 370 Step 1 · Prime Factorisation of 222 and 370 For : For : Step 2 · Calculate LCM Taking the highest power of each prime factor (, , , ): Answer > (d) 1110 Common Mistakes Lowest vs. Highest Power: Taking the lowest power of common factors calculates the HCF, not the LCM. Always take the highest power of every prime factor present. Missing Non-Common Factors: For LCM, you must include factors that appear in only one number (e.g., in part (a), in part (c)), not just the common ones.

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Question 6

Find the LCM of the following numbers:

(a) 30, 72

(b) 36, 54

(d) 222, 370

Solution

Understand the Question

To find the Least Common Multiple (LCM) of a set of numbers using the prime factorisation method:
1. Find the prime factors of each number and express them in exponential form.
2. Identify all unique prime factors across the numbers.
3. Take the highest power of each prime factor and multiply them together.

(a) 30, 72

Step 1 · Prime Factorisation of 30 and 72

For $30$ :

\begin{aligned} 30 \div 2 &= 15 \\ 15 \div 3 &= 5 \\ 5 \div 5 &= 1 \end{aligned}

$30 = 2 \times 3 \times 5$

For $72$ :

\begin{aligned} 72 \div 2 &= 36 \\ 36 \div 2 &= 18 \\ 18 \div 2 &= 9 \\ 9 \div 3 &= 3 \\ 3 \div 3 &= 1 \end{aligned}

$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$

Step 2 · Calculate LCM

Taking the highest power of each prime factor ( $2^3$ , $3^2$ , $5^1$ ):

\begin{aligned} \operatorname{LCM}(30, 72) &= 2^3 \times 3^2 \times 5^1 \\ &= 8 \times 9 \times 5 \\ &= 72 \times 5 \\ &= 360 \end{aligned}

Answer

(a) 360

(b) 36, 54

Step 1 · Prime Factorisation of 36 and 54

For $36$ :

\begin{aligned} 36 \div 2 &= 18 \\ 18 \div 2 &= 9 \\ 9 \div 3 &= 3 \\ 3 \div 3 &= 1 \end{aligned}

$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

For $54$ :

\begin{aligned} 54 \div 2 &= 27 \\ 27 \div 3 &= 9 \\ 9 \div 3 &= 3 \\ 3 \div 3 &= 1 \end{aligned}

$54 = 2 \times 3 \times 3 \times 3 = 2^1 \times 3^3$

Step 2 · Calculate LCM

Taking the highest power of each prime factor ( $2^2$ , $3^3$ ):

\begin{aligned} \operatorname{LCM}(36, 54) &= 2^2 \times 3^3 \\ &= 4 \times 27 \\ &= 108 \end{aligned}

Answer

(b) 108

(c) 105, 195, 65

Step 1 · Prime Factorisation of 105, 195, and 65

For $105$ :

\begin{aligned} 105 \div 3 &= 35 \\ 35 \div 5 &= 7 \\ 7 \div 7 &= 1 \end{aligned}

$105 = 3 \times 5 \times 7$

For $195$ :

\begin{aligned} 195 \div 3 &= 65 \\ 65 \div 5 &= 13 \\ 13 \div 13 &= 1 \end{aligned}

$195 = 3 \times 5 \times 13$

For $65$ :

\begin{aligned} 65 \div 5 &= 13 \\ 13 \div 13 &= 1 \end{aligned}

$65 = 5 \times 13$

Step 2 · Calculate LCM

Taking the highest power of each prime factor ( $3^1$ , $5^1$ , $7^1$ , $13^1$ ):

\begin{aligned} \operatorname{LCM}(105, 195, 65) &= 3^1 \times 5^1 \times 7^1 \times 13^1 \\ &= 3 \times 5 \times 7 \times 13 \\ &= 15 \times 91 \\ &= 1365 \end{aligned}

Answer

(c) 1365

(d) 222, 370

Step 1 · Prime Factorisation of 222 and 370

For $222$ :

\begin{aligned} 222 \div 2 &= 111 \\ 111 \div 3 &= 37 \\ 37 \div 37 &= 1 \end{aligned}

$222 = 2 \times 3 \times 37$

For $370$ :

\begin{aligned} 370 \div 2 &= 185 \\ 185 \div 5 &= 37 \\ 37 \div 37 &= 1 \end{aligned}

$370 = 2 \times 5 \times 37$

Step 2 · Calculate LCM

Taking the highest power of each prime factor ( $2^1$ , $3^1$ , $5^1$ , $37^1$ ):

\begin{aligned} \operatorname{LCM}(222, 370) &= 2^1 \times 3^1 \times 5^1 \times 37^1 \\ &= 2 \times 3 \times 5 \times 37 \\ &= 30 \times 37 \\ &= 1110 \end{aligned}

Answer

(d) 1110

Common Mistakes

Lowest vs. Highest Power: Taking the lowest power of common factors calculates the HCF, not the LCM. Always take the highest power of every prime factor present.
Missing Non-Common Factors: For LCM, you must include factors that appear in only one number (e.g., $5$ in part (a), $7$ in part (c)), not just the common ones.

Question 6

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