6 tiles in 3 pairs: 1 × 12, 2 × 6 and 3 × 4. So 12 has 6 factors. Students often forget 1 × 12, the pair every number has.

12 both times; the answer lands where the tile you dropped on was. Multiplication is commutative: 3 × 4 = 4 × 3.

7 gives only 1 × 7; 8 gives 1 × 8 and 2 × 4. A prime has exactly one factor pair, so exactly two factors: that is the definition of prime.

30 = 2 × 3 × 5: a blue 2, a red 3 and a green 5. Every prime up to 29 has its own colour; composite numbers are white.

6, 12, 18, 24, 30: a gap of 6 each time. The multiples of 6 are the 6 times table, and they go on for ever (the tile shows the first five).

24 = 3 × 8, then 8 = 2 × 4, then 4 = 2 × 2. You end with 3, 2, 2, 2: the prime factors, 24 = 2³ × 3. Taking out factors until every tile is prime is the same as a factor tree.

2, 3, 5, 7 and 11 are coloured: the 5 primes. 4, 6, 8, 9, 10 and 12 are white: the 6 composites. 1 is grey with a dashed edge because it is neither.

None of them is prime: 51 = 3 × 17, 57 = 3 × 19, 91 = 7 × 13. Odd numbers that don’t end in 5 look prime; adding the digits (5 + 1 = 6) shows 51 and 57 are multiples of 3.

1 has only one factor (1 × 1), while 2 and 3 each have exactly two. A prime has exactly two factors, so 1 is neither prime nor composite. Many students say 1 is prime.

Only 2 is prime. Every other even number has 1, 2 and itself as factors, so at least three factors: 2 is the only even prime.

3 primes: 41, 43 and 47 (primes from 31 up are dark grey). 49 = 7 × 7 catches people out: it is odd and doesn’t end in 5, but it isn’t prime.

12 has two blue dots and one red (2 × 2 × 3), 18 one blue and two red, 30 blue, red and green, 36 two of each. 60 has two blue, one red and one green: 60 = 2² × 3 × 5.

10 tiles in 5 pairs, but only 9 factors, because 6 × 6 uses 6 twice. Square numbers are the only numbers with an odd number of factors.

60 has 6 pairs (12 factors), 48 has 5 pairs (10 factors). 60 = 2² × 3 × 5 has three different primes to combine; 48 = 2⁴ × 3 has only two.

72 has 12 factors (6 pairs); 64 has 7 (4 pairs). 64 = 2⁶ is all one prime, while 72 = 2³ × 3² mixes two. Mixing primes gives more factors.

Each has a pair of equal numbers: 1 × 1, 2 × 2, 3 × 3 and so on. The factor counts are 1, 3, 3, 5, 3 and 9: always odd.

1 × 24, 2 × 12, 3 × 8, 4 × 6. The next pair would be 6 × 4, the same pair again. You only need to test numbers up to the square root (√24 ≈ 4.9).

There are 8: 12, 16, 18, 20, 28, 32, 44 and 45. All but 16 have 6 factors; 16 has 5 because 4 × 4 is a pair of equal numbers.

4 × 6 is the most square-like and 1 × 24 the longest and thinnest. Each factor pair is a rectangle you can make with 24 squares, so 24 makes 4 different rectangles.

⚛ gives 2, 2, 3 (12 = 2² × 3); ✂ gives 1 × 12, 2 × 6 and 3 × 4. Factor pairs are all the ways to make 12 from two numbers; prime factors are the building blocks every factor is made from.

30 = 2 × 3 × 5, 60 = 2² × 3 × 5, 120 = 2³ × 3 × 5. The 3 and 5 stay; each doubling adds one more 2.

100 makes 4 tiles (2, 2, 5, 5) and 128 makes 7 (all 2s), but 101 is prime and makes just one. The number of prime factors depends on how the number is built, not how big it is.

Yes: both end as 2, 2, 3, 5, so 60 = 2² × 3 × 5. Every number has only one prime factorisation, whatever route you take. That is the fundamental theorem of arithmetic.

8 = 2³, 27 = 3³, 125 = 5³ and 343 = 7³: each is three copies of one prime, so each tile breaks into a single colour. They are the cubes of the primes.

360 = 2³ × 3² × 5: three 2s, two 3s and one 5, so 6 tiles. The hidden tiles still show how many there are, which is a clue in itself.

The list is 2, 3, 4, 6, 9, 12, 18: every factor except 1 and 36. Taking out 1 or 36 would leave the number unchanged, so they are left out.

30 every time: (2 × 3) × 5 = 2 × (3 × 5) = 30. Multiplication is commutative and associative, which is why a prime factorisation can be written in any order.

64 = 2⁶. Pressing ⚛ Primes gives the six 2s back. Index notation counts the repeated factor: 2⁶ means six 2s multiplied, not 2 × 6.

Both make 48. 48 has 5 factor pairs (1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8) and you have made two of them.

14, 21, 28 and 35 are all composite. Any multiple of 7 bigger than 7 has 1, 7 and itself as factors, so it can’t be prime.

24 appears in both (6 × 4 and 8 × 3), so 24 is the lowest common multiple. It is not 6 × 8 = 48: 48 is a common multiple, but not the lowest.

Every multiple of 24 is a multiple of 12, because 24 = 2 × 12. Not the other way round: 36 is a multiple of 12 but not of 24.

Both trees end with 2, 2, 3 and 5: 60 = 2² × 3 × 5. Different trees always give the same primes, so students can start a tree with whichever pair they spot first.

5 ways: 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. Every tree ends with 72 = 2³ × 3².

The shortest tree has 3 levels of splits (for example 18 × 20, then 2 × 9 and 4 × 5, then 3 × 3 and 2 × 2); the longest has 5 (2 × 180, 2 × 90, 2 × 45, 3 × 15, 3 × 5). Both end 2³ × 3² × 5.

97 can’t be split: its only factor pair is 1 × 97, so it is prime and its tree is just the number. Splitting off 1 never helps, which is why 1 isn’t offered.

1000 = 2³ × 5³ (10 × 10 × 10, and 10 = 2 × 5). A common slip is 1000 = 10³, which is right but not a product of primes.

4 and 9 aren’t prime, so the tree keeps going: 36 = 2 × 2 × 3 × 3 = 2² × 3². 4 × 9 is a factorisation, but not a prime factorisation.

1296 = 2⁴ × 3⁴, so √1296 = 2² × 3² = 36. The powers in a square number’s prime factorisation are all even, and halving them gives the square root.

Both end with 2, 2, 2, 3, 5 and 7, six primes: 840 = 2³ × 3 × 5 × 7. 840 has 32 factors, the most of any number under 1000.

The middle holds 2, 3 and 5, so the HCF is 30. Everything in the diagram is 2, 2, 3, 3 and 5, so the LCM is 180.

HCF 6, LCM 36. 12 = 2² × 3 and 18 = 2 × 3², so they share one 2 and one 3.

HCF 1 and LCM 120 = 8 × 15. 8 = 2³ and 15 = 3 × 5 share no prime factors, but 1 is still a common factor, so the HCF is 1, not 0.

HCF 6, LCM 24. All of 6’s primes are in the middle, so its own circle has nothing left in it: when one number is a factor of the other, the HCF is the smaller number and the LCM the larger.

HCF 12, LCM 72 either way. Listing needs 8 factors of 24, 9 of 36 and multiples up to 72; the Venn needs only the prime factors, so it scales to big numbers.

20 × 30 = 600 and HCF × LCM = 10 × 60 = 600. It always works for two numbers: the primes in the middle are counted twice in a × b, once in the HCF and once in the LCM.

At 10:00. The LCM of 12 and 20 is 60 minutes. Students who multiply get 240 minutes (13:00), which is a common time but not the first.

1050 = 2 × 3 × 5² × 7 and 1260 = 2² × 3² × 5 × 7, so the HCF is 210 and the LCM is 6300. 1050 has 24 factors and 1260 has 36, so listing would be slow.

No: 12 has 6 factors, 24 has 8 and 48 has 10. Each doubling adds one more 2, which adds 2 factors here, not double.

2 factors: any prime. 3 factors: squares of primes only (4, 9, 25, 49). 4 factors: for example 6 or 8. 5 factors: only 16 under 50. 6 factors: for example 12.

The counts are 2, 3, 4, 5, 6 and 7: 2ⁿ has n + 1 factors (1, 2, 4, …, 2ⁿ), so 2¹⁰ = 1024 has 11.

6 (1 + 2 + 3 = 6) and 28 (1 + 2 + 4 + 7 + 14 = 28) are perfect numbers. 12 gives 16, so it is abundant. The next perfect number is 496.

24 has 8 factors and 60 has 12, the smallest of each. A number with lots of factors uses small primes, more than once.

All five have 12 factors, the most for any number under 100. Each has at least two different small primes, one of them repeated.

96 has 12 factors; 97 is prime, so it has 2. Neighbouring numbers can have very different numbers of factors.

Those are the factors of 4. The multiples are 4, 8, 12, 16, … Factors are never bigger than the number; multiples are never smaller.

All five are odd and composite: 9 = 3², 15 = 3 × 5, 21 = 3 × 7, 25 = 5² and 27 = 3³. Nearly all primes are odd, but most odd numbers aren’t prime.

The LCM is 24, not 48. Multiplying only gives the LCM when the HCF is 1; otherwise the shared primes get counted twice (here the 2).

The HCF is 9, equal to the smaller number, because 9 is a factor of 27. A number is a factor of itself.

91 = 7 × 13. Because 91 isn’t in the times tables most students know by heart, many guess it is prime.

1, 2, 4, 6, 12, 24, 36, 48 and 60 (with 1, 2, 3, 4, 6, 8, 9, 10 and 12 factors). Their prime powers never go up as the primes get bigger: 60 = 2² × 3 × 5.

17 of them: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27 and 30. The other 12 all have a prime factor of 7 or more.

8 pairs: 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73. Apart from 4, the number between is always a multiple of 6. Nobody knows whether twin primes go on for ever.

90 to 96: 7 composites in a row, between the primes 89 and 97. Gaps between primes get longer on the whole, but not steadily.

Good questions start with a prediction and have an answer the tool can check. Ask students to say which mode and buttons their partner should use.