Factors & Multiples flashcards
9 cards on Factors & Multiples: the key points, worked examples and the slips to watch. Swipe through them, or print them.
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Factors & Multiples
Factors and multiples
What is the difference between a factor and a multiple of a number?
Answer it in your head, then turn the card over.
Factors and multiples
Key points
- A factor divides into the number exactly: the factors of 12 are 1, 2, 3, 4, 6 and 12.
- A multiple is in the number's times table: 12, 24, 36 and so on.
- Factors are never bigger than the number; multiples are never smaller.
- Find factors in pairs that multiply to the number, and stop when the pairs meet.
The factors of 36
Pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
Factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36
Watch out
Listing multiples as factors: 24 and 36 are multiples of 12, not factors of it.
Factors & Multiples
Prime numbers
What makes a number prime, and how can you test whether a number is prime?
Answer it in your head, then turn the card over.
Prime numbers
Key points
- A prime has exactly two factors: 1 and itself.
- 1 is not prime: it has only one factor. 2 is the only even prime.
- The primes under 30: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
- To test a number, try dividing it by each prime up to its square root.
Is 91 prime?
is about 9.5, so try 2, 3, 5 and 7
91 ÷ 7 = 13, so 91 = 7 × 13 and is not prime
Watch out
Odd is not the same as prime: 9, 15, 21 and 91 are all odd, but none of them is prime.
Factors & Multiples
Products of prime factors
How do you write a number as a product of its prime factors?
Answer it in your head, then turn the card over.
Products of prime factors
Key points
- Split the number into any two factors, and keep splitting until every branch ends in a prime.
- Write the primes multiplied together, smallest first, using powers for repeats.
- Every number has only one prime factorisation, whichever factor tree you draw.
180 as a product of primes
180 = 2 × 90 = 2 × 2 × 45
45 = 3 × 15 = 3 × 3 × 5
So 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
Watch out
Stopping at a number that is not prime: 60 = 2 × 2 × 15 is not finished, as 15 = 3 × 5.
Factors & Multiples
HCF and LCM
What are the HCF and the LCM of two numbers, and how do you find them by listing?
Answer it in your head, then turn the card over.
HCF and LCM
Key points
- HCF, the highest common factor: the biggest number that divides into both.
- LCM, the lowest common multiple: the smallest number both divide into.
- List the factors of the smaller number for the HCF; go up the bigger number's times table for the LCM.
16 and 24
Factors of 16: 1, 2, 4, 8, 16. The biggest that divides 24 is 8, so the HCF is 8
Multiples of 24: 24, 48. 48 divides by 16, so the LCM is 48
Watch out
Multiplying the numbers does not always give the LCM: 16 × 24 = 384 is a common multiple, but not the lowest.
Factors & Multiples
Checking with HCF × LCM
How are the HCF and LCM of two numbers linked, and how can you use it?
Answer it in your head, then turn the card over.
Checking with HCF × LCM
Key points
- For two numbers, HCF × LCM = the two numbers multiplied together.
- So if you know one of them, you can find the other: LCM = product ÷ HCF.
- The product counts the shared factors twice, which is why it is often bigger than the LCM.
12 and 18
The HCF is 6, so the LCM = 12 × 18 ÷ 6 = 36
Check: 6 × 36 = 216, and 12 × 18 = 216
Watch out
This link is for two numbers only: it does not hold for the HCF and LCM of three numbers.
Factors & Multiples
HCF and LCM from a Venn diagram
How do you use a Venn diagram of prime factors to find the HCF and the LCM?
Answer it in your head, then turn the card over.
HCF and LCM from a Venn diagram
Key points
- Write each number as a product of primes.
- Put the primes they share in the overlap, and the rest in each number's own part.
- HCF = the product of the overlap. LCM = the product of every prime in the diagram.
84 = 2² × 3 × 7 and 90 = 2 × 3² × 5
Overlap: 2 and 3, so the HCF = 2 × 3 = 6
84 only: 2 and 7. 90 only: 3 and 5
LCM = 2 × 7 × 2 × 3 × 3 × 5 = 1260
Watch out
The HCF uses only the overlap: multiplying every prime in the diagram gives the LCM.
Factors & Multiples
HCF and LCM from index form
How do you read the HCF and the LCM straight from two prime factorisations written with powers?
Answer it in your head, then turn the card over.
HCF and LCM from index form
Key points
- HCF: take each prime that is in both numbers, at the lower of its two powers.
- LCM: take every prime that is in either number, at the higher power.
- Leave the answer in index form unless the question asks for its value.
2³ × 3 × 5² and 2 × 3² × 5
HCF: the lower powers, 2 × 3 × 5 = 30
LCM: the higher powers, 2³ × 3² × 5² = 1800
Watch out
Swapping the powers: the HCF takes the lower power of each shared prime, the LCM the higher.
Factors & Multiples
When things next happen together
Which do you need when a problem asks when two things will next happen at the same time?
Answer it in your head, then turn the card over.
When things next happen together
Key points
- "When do they next happen together?" is an LCM question.
- Find the LCM of the two times, then add it on to the start time.
- Give the answer as the problem asks: a time on the clock, a day, a number of laps.
Trains every 12 and 18 minutes
Both leave at 7:40. Multiples of 18: 18, 36. 36 divides by 12, so the LCM is 36
40 + 36 = 76 minutes past 7, which is 8:16
Watch out
Using the HCF for "next together": buses every 12 and 20 minutes meet after 60 minutes, not 4.
Factors & Multiples
Sharing into the biggest equal groups
Which do you need when a problem asks for the largest number of identical groups, or the biggest size that fits exactly?
Answer it in your head, then turn the card over.
Sharing into the biggest equal groups
Key points
- "The most identical groups" or "the biggest equal pieces" is an HCF question.
- The number of groups has to divide every amount exactly.
- Then divide each amount by the HCF to say what goes in each group.
30 apples and 42 pears in identical bags
The HCF of 30 and 42 is 6, so 6 bags
Each bag holds 30 ÷ 6 = 5 apples and 42 ÷ 6 = 7 pears
Watch out
Using the LCM for equal groups: 30 apples and 42 pears could never fill 210 identical bags.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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