Cheat sheet: Number properties
Factors and multiples
- Factors divide exactly into a number: factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- Multiples are in its times table: 4, 8, 12, 16 ...
- Factors come in pairs.
Common factors and multiples
- Common factors of 16 and 40: 1, 2, 4, 8. Highest (HCF) = 8.
- Common multiples of 3 and 5: 15, 30, ... Lowest (LCM) = 15.
Prime numbers
Exactly 2 factors: 1 and itself. 1 is not prime. 2 is the only even prime.
Under 20: 2, 3, 5, 7, 11, 13, 17, 19. There are 25 under 100.
Squares and cubes
- Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Cubes: 1, 8, 27, 64, 125.
- 5² = 5 × 5 = 25, not 10.
Divisibility tests
- 2: ends in 0, 2, 4, 6, 8.
- 3: digits add to a multiple of 3.
- 4: last two digits divide by 4.
- 5: ends in 0 or 5. 10: ends in 0.
- 6: even and divides by 3.
- 9: digits add to a multiple of 9.
Prime factors
Factor tree: keep splitting until every end is prime. 90 = 2 × 3 × 3 × 5.
HCF: multiply the primes both numbers have, as many times as both have them. 12 = 2×2×3 and 8 = 2×2×2 share 2×2, so the HCF is 4.
LCM: for each prime, take the most times it appears in either number, then multiply. 12 and 8: 2×2×2×3 = 24.
“Who am I?” puzzles
List the numbers that fit the first clue. Test each one on the next clue. Cross out until one is left.
Words to know
factor, multiple, prime, composite (a whole number bigger than 1 that is not prime, e.g. 4, 6, 9), square, cube, product, divisible, HCF, LCM, odd, even.
Mini lessons
Factor pairs and common factors
Skills: PROP-01, PROP-03
A factor is a number that divides exactly into another. Factors come in pairs that multiply to make the number. A common factor of two numbers is a factor of both of them.
Join the smallest factor to the biggest, and work inwards. Each join is a factor pair.
List all the factors of 42. Then find the common factors of 16 and 40 and the highest one.
- Pairs for 42: 1 × 42, 2 × 21, 3 × 14, 6 × 7.
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
- Factors of 16: 1, 2, 4, 8, 16. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
- In both lists: 1, 2, 4 and 8. The highest is 8.
Answer: Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Common factors of 16 and 40: 1, 2, 4, 8, and the highest is 8
Multiples and common multiples
Skills: PROP-02
Multiples are the numbers in a times table. A common multiple is in the times tables of both numbers. The lowest common multiple is the first time the two lists meet.
15 is the only number up to 24 in both lists. The first common multiple of 3 and 5 is 15.
One bus leaves every 6 minutes and another every 8 minutes. They leave together at 9:00. When do they next leave together?
- Multiples of 6: 6, 12, 18, 24, 30 ...
- Multiples of 8: 8, 16, 24, 32 ...
- The first number in both lists is 24.
- 24 minutes after 9:00 is 9:24.
Answer: 9:24
Prime numbers
Skills: PROP-04
A prime number has exactly two factors: 1 and itself. The number 1 is not prime because it has only one factor. To find the primes, cross out 1. Keep 2, 3, 5 and 7, but cross out all their other multiples. What is left up to 100 is prime.
Is 51 prime? Is 97 prime?
- Test 51: the digits add to 5 + 1 = 6, so it divides by 3.
- 51 = 3 × 17, so it has more than two factors. It is not prime.
- Test 97: not even, does not end in 5, digits add to 16 so not in the 3 table.
- 97 ÷ 7 = 13 remainder 6, so 7 does not work either. No test works, so 97 is prime.
Answer: 51 is not prime; 97 is prime
Square and cube numbers
Skills: PROP-05
A square number is a number multiplied by itself, such as 6 × 6 = 36, written 6². A cube number is a number multiplied by itself three times, such as 3 × 3 × 3 = 27, written 3³.
Square numbers make a square of dots. 4 × 4 = 4² = 16. A cube number is a × a × a, such as 3 × 3 × 3 = 27.
Which numbers under 100 are both square and cube numbers?
- Cubes under 100: 1, 8, 27, 64.
- Squares under 100: 1, 4, 9, 16, 25, 36, 49, 64, 81.
- Numbers in both lists: 1 and 64.
- Check: 64 = 8 × 8 and 64 = 4 × 4 × 4.
Answer: 1 and 64
Divisibility, odd and even
Skills: PROP-06
Quick tests tell you if a number divides exactly without doing a long division. Even numbers end in 0, 2, 4, 6 or 8 and odd numbers end in 1, 3, 5, 7 or 9. For 3 and 9, add the digits.
If the digit total is in the 3 times table, the number divides by 3. In the 9 times table: it divides by 9.
Is 4,716 divisible by 3? By 4? By 6? By 9?
- Digits: 4 + 7 + 1 + 6 = 18. 18 is in the 3 times table and the 9 times table, so yes for 3 and yes for 9.
- Last two digits: 16. 16 ÷ 4 = 4, so yes for 4.
- Even and divisible by 3, so yes for 6.
Answer: Yes to all four
“Who am I?” number puzzles
Skills: PROP-07
Each clue narrows the list. Start with the clue that gives the shortest list, then test each number against the other clues.
Clues cut down the list until one number is left.
I am a prime number between 30 and 40. My digits add up to 10. Who am I?
- Primes between 30 and 40: 31 and 37.
- Digit total of 31: 3 + 1 = 4.
- Digit total of 37: 3 + 7 = 10.
Answer: 37
Prime factors, HCF and LCM
Skills: PROP-08. Stretch: usually taught in Year 7, but some 11+ tests include it.
Every whole number above 1 can be written as a product of primes. A factor tree shows how. Once you have the prime factors, the HCF (highest common factor) and LCM (lowest common multiple) are quick to find.
Write 20 and 30 as products of primes. Then find their HCF and LCM.
- 20 = 2 × 2 × 5 and 30 = 2 × 3 × 5.
- HCF: multiply the primes they share: 2 × 5 = 10.
- LCM: use each prime the most times it appears in either number: 2 × 2 × 3 × 5 = 60.
- Check: 10 goes into both 20 and 30, and 60 is in both the 20 and 30 times tables (20 × 3 and 30 × 2).
Answer: HCF = 10, LCM = 60
Worked examples across the levels
Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.
Level 1 (Year 3): one step
Write the first five multiples of 4.
- Count in 4s: 4, 8, 12, 16, 20.
Answer: 4, 8, 12, 16, 20
Level 2 (Year 4): one or two steps
Write all the factors of 20.
- Pairs: 1 × 20, 2 × 10, 4 × 5.
- List them in order.
Answer: 1, 2, 4, 5, 10, 20
Level 3 (Year 5): two steps
Find the common factors of 24 and 36, and say which is the highest.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- In both: 1, 2, 3, 4, 6, 12.
Answer: 1, 2, 3, 4, 6, 12; the highest is 12
Level 4 (exam standard): several steps or working backwards
One light flashes every 12 seconds and another every 15 seconds. They flash together at 9:00:00. When do they next flash together, and how many times has the 15-second light flashed after 9:00:00, up to and including that flash?
- Multiples of 12: 12, 24, 36, 48, 60.
- Multiples of 15: 15, 30, 45, 60.
- First in both: 60 seconds, so 9:01:00.
- The 15-second light flashed at 15, 30, 45 and 60: 4 times (60 ÷ 15 = 4).
Answer: 9:01:00, and 4 flashes
Level 5 (challenge): three to five steps
I am a two-digit number. I am a multiple of 4 and a multiple of 6. I am not a multiple of 8. My digits add up to a square number. What number am I?
- Multiples of both 4 and 6 are multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96.
- Remove multiples of 8: 24, 48, 72, 96 go. Left: 12, 36, 60, 84.
- Digit totals: 3, 9, 6, 12.
- Only 9 is a square number (3 × 3), so the number is 36.
Answer: 36
Common mistakes
Mixing up factors and multiples
The mistake: Writing 3, 6, 9, 12 when asked for the factors of 12.
Why it is wrong: Those are multiples of 3. Factors of 12 are the numbers that divide into 12.
How to avoid it: Factors are Few and small (they go into the number). Multiples are Many and keep growing.
Thinking 1 is prime, or that all odd numbers are prime
The mistake: Saying 1 is prime, or that 9 and 15 are prime because they are odd.
Why it is wrong: A prime has exactly two factors. 1 has only one. 9 = 3 × 3 and 15 = 3 × 5 have more.
How to avoid it: Test with the divisibility rules for 2, 3, 5 and 7 before you decide.
Squaring means doubling
The mistake: Writing 5² = 10.
Why it is wrong: 5² means 5 × 5, not 5 × 2.
How to avoid it: Say it as “5 times 5”. Picture a 5 by 5 square of dots.
Missing factors in a pair
The mistake: Listing the factors of 28 as 1, 2, 4, 7, 14 and forgetting 28.
Why it is wrong: Every number is a factor of itself. Without pairs, it is easy to leave out the big ones.
How to avoid it: Work in pairs: 1 × 28, 2 × 14, 4 × 7. Write both numbers from each pair.
Quick check
Five questions, easy to hard. Use paper and have a go before you open the answer.
Which of these is a factor of 45: 4, 6, 9 or 12?
Show answer
9
9 × 5 = 45. The others do not divide into 45 exactly.
How many prime numbers are there between 10 and 20?
Show answer
4
The primes are 11, 13, 17 and 19. (15 divides by 5, 12, 14, 16, 18 are even.)
What is 4² + 3³?
Show answer
43
4² = 16 and 3³ = 27. 16 + 27 = 43.
A bus leaves every 6 minutes and a tram every 9 minutes. They both leave at 8:00. When do they next leave together?
Show answer
8:18
Multiples of 6: 6, 12, 18. Multiples of 9: 9, 18. The first in both is 18 minutes.
Write 84 as a product of prime numbers.
Show answer
2 × 2 × 3 × 7
84 = 4 × 21 = 2 × 2 × 3 × 7. Check: 2 × 2 = 4, 4 × 3 = 12, 12 × 7 = 84.
For grown-ups
- Practise times tables both ways. If your child knows 6 × 7 = 42, ask “what are the factors of 42?” Knowing tables well makes this whole topic easier.
- Play “Who am I?” at the dinner table: “I am a multiple of 7 between 40 and 60”. Take turns writing the clues.
- Use number plates, house numbers or page numbers for quick checks: “is it divisible by 3?” “is it prime?”