LCM • Topic 1 of 3

LCM by Factorization

Write each number as a product of primes, then take every prime that appears in any number, each raised to its HIGHEST power across the set — multiply those together and you have the LCM. For 12 = 2²×3 and 18 = 2×3², the LCM uses 2² (highest power of 2) and 3² (highest power of 3), giving 4×9 = 36. Contrast this with HCF, which takes the LOWEST power of only the COMMON primes (here 2¹×3¹ = 6). The factor method is faster and far less error-prone than the school "ladder" division when the numbers are coprime or have large factors, and it is the only safe method when CAT mixes in powers like 2⁴ or 5³. A clean check: the LCM must be divisible by every number in the set, and it can never be smaller than the largest number given.

✅ Solved examples

1. Find the LCM of 12, 15 and 20.
12 = 2²×3, 15 = 3×5, 20 = 2²×5. Highest powers: 2²×3×5 = 4×3×5 = 60.
2. Find the LCM of 8, 9 and 25.
All coprime: 8 = 2³, 9 = 3², 25 = 5². LCM = 2³×3²×5² = 8×9×25 = 1800.
3. Find the LCM of 2³×3², 2²×3³×5 and 2×5².
Highest of each prime: 2³, 3³, 5². LCM = 8×27×25 = 5400.
4. Find the LCM of 36, 48 and 72.
36 = 2²×3², 48 = 2⁴×3, 72 = 2³×3². Highest: 2⁴×3² = 16×9 = 144.

✏️ Practice — try these, take hints as needed

1. Find the LCM of 10, 15 and 25.
Factorise each.
10 = 2×5, 15 = 3×5, 25 = 5².
Take 2, 3 and 5² (highest power of 5).
150
2. Find the LCM of 14, 21 and 35.
Each has 7 as a factor.
14 = 2×7, 21 = 3×7, 35 = 5×7.
LCM = 2×3×5×7.
210
3. Find the LCM of 16, 24 and 40.
Powers of 2 dominate.
16 = 2⁴, 24 = 2³×3, 40 = 2³×5.
Take 2⁴×3×5.
240
4. Find the LCM of 9, 12 and 15.
9 = 3², 12 = 2²×3, 15 = 3×5.
Highest power of 3 is 3².
2²×3²×5.
180
5. Find the LCM of 2²×5, 2³×3 and 3²×5.
List primes 2, 3, 5.
Highest powers: 2³, 3², 5¹.
8×9×5.
360

📝 Topic test — 8 questions

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HCF-LCM Relationship →