50 = 2^1 × 5^2 and 20 = 2^2 × 5^1. Their L.C.M. = 2^□ × 5^□ Fill in each blank

2, 2. For the L.C.M., take each prime's larger index: 2 to the 2, 5 to the 2.

The H.C.F. of 12 and 54 is 6. True or false?

True. 12 = 6 × 2 and 54 = 6 × 9: the H.C.F. is 6. A smaller common factor isn't the highest.

360 = 2^3 × 3^2 × 5 and 42 = 2^1 × 3^1 × 7. Their H.C.F. is □ and their L.C.M. is □. Fill in each blank

6, 2520. 5 and 7 each belong to one number only: not common, so not in the H.C.F., though the L.C.M. takes them. The H.C.F. takes 2 and 3 to the smaller index: 6. The L.C.M. takes every prime to the larger: 2520.

Working with numbers · Ages 12–13

H.C.F. and L.C.M.

H.C.F.: smaller indices. L.C.M.: larger indices.

How to do it

Method

H.C.F.: smaller indices. L.C.M.: larger indices

12 = 2² × 3, 18 = 2 × 3²: H.C.F. = 6, L.C.M. = 36

Another way

Dividing both at once

12, 18 ÷ 2 ÷ 3 → 2, 3: H.C.F. = 6, L.C.M. = 6 × 2 × 3 = 36

In full

Break both numbers into primes first. The highest common factor (H.C.F.) takes the primes both have, each to the smaller index; the lowest common multiple (L.C.M.) takes every prime that appears, each to the larger index. 12 = 2² × 3 and 18 = 2 × 3²: H.C.F. = 2 × 3 = 6, L.C.M. = 2² × 3² = 36. Or divide both by common factors at once: 2 and 3 take 12 and 18 down to 2 and 3, so H.C.F. = 2 × 3 = 6 and L.C.M. = 6 × 2 × 3 = 36. For two numbers, their product = H.C.F. × L.C.M.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
50 = 2¹ × 5² and 20 = 2² × 5¹. Their L.C.M. = 2^□ × 5^□
Fill in each blank
Show the answer and working
Answer: 2, 2
For the L.C.M., take each prime's larger index: 2 to the 2, 5 to the 2.
Example 2
The H.C.F. of 12 and 54 is 6.
True or false?
Show the answer and working
Answer: True
12 = 6 × 2 and 54 = 6 × 9: the H.C.F. is 6. A smaller common factor isn't the highest.
Example 3
360 = 2³ × 3² × 5 and 42 = 2¹ × 3¹ × 7. Their H.C.F. is and their L.C.M. is .
Fill in each blank
Show the answer and working
Answer: 6, 2520
5 and 7 each belong to one number only: not common, so not in the H.C.F., though the L.C.M. takes them. The H.C.F. takes 2 and 3 to the smaller index: 6. The L.C.M. takes every prime to the larger: 2520.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.The lowest common multiple (L.C.M.) of 88 and 99 is .
Answer:
2.The highest common factor (H.C.F.) of 9 and 12 is .
Answer:
3.200 = 2³ × 5² and 400 = 2⁴ × 5². Their L.C.M. = 2^□ × 5^□
Fill in each blank
Answer:
4.48 = 2⁴ × 3¹ and 432 = 2⁴ × 3³. Their H.C.F. = 2^□ × 3^□
Fill in each blank
Answer:
5.What is the L.C.M. of 6 and 8?
- A. 24
- B. 14
- C. 2
- D. 48
Choose the right answer
Answer:

All the answers

1. 792
2. 3
3. 4, 2
4. 4, 1
5. A. 24

Free worksheet: H.C.F. and L.C.M.Working out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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