Common division procedure for HCF and LCM

The idea

Divide both numbers by a common prime factor, writing the quotients below, and repeat until the quotients have no common prime factor; the HCF is the product of the divisors, and the LCM is that product times the final quotients.

Dividing together

84 and 180 are both even. Divide both by 2: we get 42 and 90.

42 and 90 are still both even. What would you do next?

• Divide both by 2 again
• Stop, we are done
• Divide only 42

Climb down the ladder

Divide both numbers by a common prime and write the two answers below. Keep going until they share no prime. Then pick the numbers that make the HCF, and then the LCM.

The ladder holds both answers

Divide both numbers by a common prime, write the two quotients in the next row, and repeat until the quotients have no common prime factor.

HCF = the product of the divisors = 2 × 2 × 3 = 12. LCM = the divisors times the last quotients = 2 × 2 × 3 × 7 × 15 = 1260.

Why? Every divisor went into both numbers, so together they are the common part (the HCF). The last quotients, 7 and 15, are what is left over in each number. Common part and both leftovers make the smallest number that holds both.

Divide both numbers by a common prime factor, writing the quotients below, and repeat until the quotients have no common prime factor. The HCF is the product of the divisors. The LCM is that product times the final quotients.

• Divide both by | 84 and 180
• 84, 180
• 2 | 42, 90
• 2 | 21, 45
• 3 | 7, 15

Notes

Divide both numbers by a common prime, writing the quotients below, until the quotients have no common prime factor. HCF = product of the divisors. LCM = that product × the final quotients.

Check yourself

A ladder for 60 and 90 divides by 2, then 3, then 5. The last quotients are 2 and 3. What is the HCF?

Answer: 30

2 × 3 × 5 = 30. The numbers go 60, 90 → 30, 45 → 10, 15 → 2, 3, and 2 and 3 share no prime.

A ladder for 36 and 48 divides by 2, 2 and 3 and ends with the quotients 3 and 4. What is the LCM?

Answer: 144

HCF = 2 × 2 × 3 = 12. LCM = 12 × 3 × 4 = 144. And 144 = 36 × 4 = 48 × 3.

The ladder of two numbers has reached one of these pairs. For which pair do we stop?

In the ladder for 36 and 48, the divisors are 2, 2, 3 and the last quotients are 3 and 4. Why is 2 × 2 × 3 × 3 × 4 a common multiple of 36 and 48?

• 12 and 18. Both are divisible by 2 (and 3, and 6), so they still share a prime. Keep dividing.
• 10 and 25. Both are divisible by 5. Divide by 5 once more.
• 14 and 15 — correct. Yes! 14 = 2 × 7 and 15 = 3 × 5 share no prime. Stop.
• It is 36 × 4 and also 48 × 3: it holds all of both numbers. — correct. Yes! The product is 12 × 3 × 4 = 144. 36 = 12 × 3 and 48 = 12 × 4, so each number fits inside it.
• Because we multiplied every number we wrote down.. The reason is not the number of numbers. We must be sure each of 36 and 48 sits inside the product.
• Because 2 × 2 × 3 is the HCF.. The HCF is only the common part. The LCM also needs the leftovers 3 and 4.