Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Which number is divisible by both 4 and 9?
Show worked solution
For 9: 3472 has digit sum 16, so it fails. For 4: 6318 ends in 18 and 7542 ends in 42, neither a multiple of 4, so both fail. 5724 has digit sum 18 and ends in 24 = 4 × 6. Check: 5724 = 36 × 159.
2 Find the HCF and the LCM of 84 and 120.
Show worked solution
84 = 22 × 3 × 7 and 120 = 23 × 3 × 5. HCF = 22 × 3 = 12. LCM = 23 × 3 × 5 × 7 = 840. Check: 12 × 840 = 10 080 = 84 × 120.
3 How many whole numbers from 1 to 100 are multiples of 6 but NOT multiples of 4?
Show worked solution
Multiples of 6: 100 ÷ 6 = 16 remainder 4, so 16 of them. Those that are also multiples of 4 are multiples of LCM 12: 100 ÷ 12 = 8 remainder 4, so 8 of them. 16 − 8 = 8.
Level 2 · Core
4 What is the units digit of 132030 × 172030?
Show worked solution
32030 × 72030 = 212030, and every power of a number ending in 1 ends in 1: 1. Or: 2030 = 4 × 507 + 2, so 32030 and 72030 both end in 9, and 9 × 9 = 81. 9 is the units digit of one factor only; 8 adds 9 + 9 instead of multiplying; 3 is just the units digit of 13.
5 Find the remainder when 350 is divided by 5, and when 350 is divided by 4.
Show worked solution
Divided by 5, the remainders of 3, 9, 27, 81, … are 3, 4, 2, 1, repeating every 4. 50 = 4 × 12 + 2, so the 2nd place: 4. Divided by 4, the remainders are 3, 1, 3, 1, …, repeating every 2. 50 is even, so the 2nd place: 1.
6 N is the smallest positive whole number whose digits add up to 30. What is N?
Show worked solution
Three digits add to at most 9 + 9 + 9 = 27, so N needs 4 digits. To keep N small, make the first digit as small as possible and the rest 9s: 30 − 27 = 3, so N = 3999.
Level 3 · Challenge
7 Three statements about the number 123 456:
1 It is divisible by 3.
2 It is divisible by 8.
3 It is divisible by 11.
Which statements are correct?
1 It is divisible by 3.
2 It is divisible by 8.
3 It is divisible by 11.
Which statements are correct?
Show worked solution
Digit sum 1 + 2 + 3 + 4 + 5 + 6 = 21, a multiple of 3 (1 ✓). Last three digits 456 = 8 × 57 (2 ✓). Alternating sum 1 − 2 + 3 − 4 + 5 − 6 = −3, not a multiple of 11 (3 ✗). 1 and 2.
8 Of the whole numbers from 1 to 300, how many are divisible by 2, 3 or 5, and how many are divisible by none of them?
Show worked solution
Multiples of 2, 3, 5: 150 + 100 + 60 = 310. Pairs: multiples of 6, 10, 15 number 50 + 30 + 20 = 100, so 310 − 100 = 210. The 10 multiples of 30 were added three times and taken away three times, so add them back: 220. None: 300 − 220 = 80.
9 N is the smallest positive whole number that leaves remainder 1 when divided by 2, 3, 4, 5 or 6, and is divisible by 7. Find N.
Show worked solution
N − 1 is a multiple of LCM(2, 3, 4, 5, 6) = 60, so N is 61, 121, 181, 241, 301, … Test for 7: 61 = 56 + 5, 121 = 119 + 2, 181 = 175 + 6, 241 = 238 + 3, 301 = 7 × 43. So N = 301.