Unit 3 · Topic 02 · Playing with Numbers
Ms. Rao asked for the LCM and HCF of 8 and 20 — without long division.
Kabir started listing multiples. Anaya broke each number into prime factors instead.
By the time Kabir reached 40 on his list, Anaya had already found both answers.
"Find the LCM and HCF of 8 and 20 using prime factorisation," said Ms. Rao.
Kabir wrote the prime factors. 8 = 2 × 2 × 2. 20 = 2 × 2 × 5.
"For the LCM, line up the common prime pairs," said Ms. Rao. "Write each common pair only once, then add whatever is left over."
Anaya matched the two pairs of 2, wrote those once, then added the remaining 2 and 5. "LCM = 2 × 2 × 2 × 5 = 40. Forty is the first number that appears in both the 8-times and 20-times tables."
"And the HCF?" asked Kabir.
"Only the common pairs count," said Ms. Rao. "One pair of 2, another pair of 2 — that gives 4. Four is the largest number that divides both 8 and 20 exactly."
They tried a bigger pair next: 60 and 90. 60 = 2 × 2 × 3 × 5. 90 = 2 × 3 × 3 × 5.
For the LCM they wrote the common 2, 3 and 5 once, then added the extra 2 and the extra 3. LCM = 180.
For the HCF they kept only the common pairs: 2 × 3 × 5 = 30.
Then came 15 and 45. 15 = 3 × 5. 45 = 3 × 3 × 5. Here 15 is a factor of 45 itself.
"When one number divides the other," said Ms. Rao, "the LCM is always the larger number and the HCF is always the smaller." LCM = 45, HCF = 15.
Last example: 14 and 25. 14 = 2 × 7. 25 = 5 × 5. No common prime pair at all.
"These are co-prime," said Ms. Rao. "Their only common factor is 1, so HCF = 1 and LCM = 2 × 5 × 5 × 7 = 350 — just multiply the two numbers together."
Prime factorisation breaks a number into a product of prime numbers, often shown with a factor tree.
To find the LCM: write the prime factors of both numbers, match the common prime pairs, write each common pair once, then multiply in any primes that are left over.
To find the HCF: use only the common prime pairs — take one copy of each matched pair and multiply.
Special cases from the lesson: if one number is a factor of the other, LCM equals the larger number and HCF equals the smaller. If two numbers are co-prime (HCF = 1), LCM equals their product.
Work through 8 and 20 exactly as in the lesson video: write the prime factors, find LCM = 40 and HCF = 4.
Try 14 and 25 next — notice there are no common pairs, so HCF = 1 and LCM = 350.
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