Is 7311 divisible by 9? Its digit sum is □, and the digit sum leaves □ when divided by 9. Fill in each blank

12, 3. 7 + 3 + 1 + 1 = 12, and 12 ÷ 9 leaves 3. Only a remainder of 0 means divisible by 9.

6 = 2^1 × 3^1 and 48 = 2^4 × 3^1. Their H.C.F. = 2^□ × 3^□ Fill in each blank

1, 1. For the H.C.F., take each prime's smaller index: 2 to the 1, 3 to the 1.

1, 5. 0.2 = 1/5: write the decimal over 10, 100 or 1000, then simplify.

Numbers and operations · Ages 12–13

From “Divisibility by 4, 6, 8 and 9” to “Fractions and decimals mixed”, step by step, each skill with its methods, worked examples and answers.

Divisibility by 4, 6, 8 and 94: last two digits. 8: last three. 9: digit sum. 6: both 2 and 3
Method7236: 36 = 4 × 9, 7 + 2 + 3 + 6 = 18
Another way: Just divide7236 ÷ 4 = 1809
Powersaⁿ is n lots of a multiplied, not a × n
Method2⁵ = 2 × 2 × 2 × 2 × 2 = 32
Another way: Double again and again2 → 4 → 8 → 16 → 32
Prime factorisationDivide by 2 first, again and again, until you reach 1
Method360 = 2³ × 3² × 5
Another way: Factor tree360 = 36 × 10 = 6 × 6 × 2 × 5
H.C.F. and L.C.M.H.C.F.: smaller indices. L.C.M.: larger indices
Method12 = 2² × 3, 18 = 2 × 3²: H.C.F. = 6, L.C.M. = 36
Another way: Dividing both at once12, 18 ÷ 2 ÷ 3 → 2, 3: H.C.F. = 6, L.C.M. = 6 × 2 × 3 = 36
Brackets inside bracketsInside out: ( ) → [ ] → ｛｝
Method2 × [10 − (3 + 1)] = 2 × [10 − 4] = 2 × 6 = 12
Another way: Opening the brackets20 − (5 + 3) = 20 − 5 − 3 = 12
Fractions and decimals mixedOne form for all first, then times before plus
Method0.5 + 34 × 2 = 12 + 32 = 2
Another way: All as decimals0.5 + 0.75 × 2 = 0.5 + 1.5 = 2

From different skills in this topic. Have a go first, then open each one for the answer and the working.

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