HCF and LCM

Q: Is there a shortcut for the LCM of two numbers?

Yes: \text{LCM} = \dfrac{a \times b}{\text{HCF}}. For 12 and 18 that is 216 \div 6 = 36.

The HCF (highest common factor) is the largest number that divides two or more numbers exactly. The LCM (lowest common multiple) is the smallest number that all of them divide into. Prime factorisation finds both quickly.

Formulas

Prime factorisation method

HCF = product of the common primes, lowest powers

LCM = product of all primes, highest powers

For two numbers a and b

HCF (a, b) \times LCM (a, b) = a \times b

Worked examples

Example 1. Find the HCF and LCM of 12 and 18.

$12 = 2^{2} \times 3$ and $18 = 2 \times 3^{2}$ .
HCF: lowest powers of the common primes, $2 \times 3 = 6$ .
LCM: highest powers of all primes, $2^{2} \times 3^{2} = 36$ .
Check: $6 \times 36 = 216 = 12 \times 18$ .

Answer: HCF = 6, LCM = 36

Example 2. The HCF of two numbers is 4 and their LCM is 120. One number is 24. Find the other.

HCF × LCM = product of the numbers, so $4 \times 120 = 24 \times x$ .
$x = 480 \div 24 = 20$ .

Answer: 20

Example 3. Two bells ring every 12 minutes and every 18 minutes. They ring together at 8:00 a.m. When do they next ring together?

They coincide every LCM(12, 18) minutes.
LCM = 36 minutes, so the next time is 8:36 a.m.

Answer: 8:36 a.m.

Common mistakes

Mixing up HCF and LCM. The HCF is never bigger than the smaller number; the LCM is never smaller than the larger one.
Using highest powers for the HCF. The HCF takes the lowest power of each common prime.
Using HCF × LCM = product for three numbers. That rule works for two numbers only.

Practice questions with answers

1. What is the HCF of 12 and 18?

Show answer

Answer: A) 6

Factors of 12: 1,2,3,4,6,12. Factors of 18: 1,2,3,6,9,18. Common factors: 1,2,3,6. The highest is 6.

2. Three bells in a school ring at intervals of 9, 12 and 15 minutes.
They all ring together at 9:00 am.
How many times in the next 3 hours (from 9:00 am up to and including 12:00 noon) will they ring together again?

Show answer

Answer: A) 1

They ring together every LCM(9,12,15) minutes.

9 = 3^{2}, 12 = 2^{2} x 3, 15 = 3 \times 5

, LCM =

2^{2} x 3^{2} x 5 = 180

minutes = 3 hours. In the next 3 hours they ring together once, exactly at 12:00 noon.

3. Two numbers have HCF 11 and LCM 1694, and both numbers lie between 100 and 200.
Which pair is it?

A11 and 1694
B132 and 143
C121 and 154
D110 and 154

Show answer

Answer: C) 121 and 154

Write the numbers as

11 a

and

11 b

with a,

b

co-prime; LCM =

11 ab = 1694

gives

ab = 154 = 2 \times 7 \times

11. For both numbers to lie between 100 and 200, a and

b

must lie between 10 and 18, and (a, b) = (11, 14) works, giving 121 and 154. The pair 11 and 1694 does have HCF 11 and LCM 1694 but lies outside the stated range; 132 and 143 give LCM 1716; 110 and 154 give HCF 22, not 11.

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Frequently asked questions

What is the difference between HCF and LCM?

The HCF is the biggest number that divides all the given numbers. The LCM is the smallest number that all of them divide into.

When do I use the LCM and when the HCF in word problems?

Use the LCM for "when will they happen together again" problems (bells, traffic lights, buses). Use the HCF for "largest equal groups or pieces" problems (cutting ropes, making equal rows).

What is the HCF of two prime numbers?

It is 1, because two different primes share no common factor other than 1.

Is there a shortcut for the LCM of two numbers?

Yes: $LCM = \frac{a \times b}{HCF}$ . For 12 and 18 that is $216 \div 6 = 36$ .

HCF and LCM

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics