Long multiplication, primes, powers & roots
Long multiplication; long/short/decimal division with 2-digit divisors; factors, multiples, HCF/LCM; prime & composite numbers, prime factorisation; square & cube numbers, powers with natural exponents, powers of ten, roots & reciprocals; order of operations.
This concept gathers the tools for taking numbers apart and building them back up: multiplying and dividing big numbers, splitting a number into its prime “atoms”, and raising numbers to powers or pulling out their roots.
What it means
By now you can multiply and divide small numbers. Here we scale that up and go deeper.
Long multiplication multiplies numbers with several digits by breaking the work into place-value pieces. To do 24 × 36, you split 36 into 30 + 6, multiply 24 by each part, then add. Long division does the reverse for a 2-digit divisor — sharing a big number into equal groups, digit by digit, tracking the remainder — and can continue past the decimal point to divide decimals. (Short division is the same process for a single-digit divisor, carrying the remainder in your head instead of writing it out.)
Every whole number is built from smaller ones. A factor of a number divides it exactly (no remainder): 1, 2, 3, 4, 6, 12 are the factors of 12. A multiple is what you get counting up in steps: 12, 24, 36… are multiples of 12. The HCF (highest common factor) is the biggest factor two numbers share; the LCM (lowest common multiple) is the smallest multiple they share.
A prime number has exactly two factors — 1 and itself (2, 3, 5, 7, 11…). A composite number has more. The remarkable fact — the fundamental theorem of arithmetic — is that every whole number above 1 is a product of primes in exactly one way. This prime factorisation is a number’s fingerprint: 12 = 2 × 2 × 3.
Powers are repeated multiplication. 2⁴ means 2 × 2 × 2 × 2 = 16; the small raised number (the exponent) counts the factors. 5² = 25 is a square number, 4³ = 64 a cube number. Powers of ten (10, 100, 1000…) shift place value. A root undoes a power: √25 = 5, ∛64 = 4. A reciprocal is 1 divided by a number (the reciprocal of 4 is ¼), and a number times its reciprocal gives 1.
Finally, order of operations fixes which step comes first when several are mixed: brackets, then powers/roots, then × and ÷, then + and −. So 2 + 3 × 4 = 14, not 20.
Worked examples
Long multiplication — 24 × 36:
24
× 36
----
144 (24 × 6)
720 (24 × 30)
----
864Prime factorisation with a factor tree — 60:
60
/ \
6 10
/ \ / \
2 3 2 5So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
HCF and LCM from the factorisations of 12 and 18:
12 = 2² × 3,18 = 2 × 3²- HCF = shared factors =
2 × 3 = 6 - LCM =
2² × 3² = 36
Powers and roots: 3² = 9, 2³ = 8, √49 = 7, and 10³ = 1000.
The generative-art connection
Multiplication is literally a picture: 4 × 6 is a rectangle of dots, 4 rows by 6 columns — an array. The dot-multiplier tool turns one dot into a symmetric starburst, so counting and multiplication become something you watch grow.
Times tables become curves. Place 36 points evenly around a circle, then join each point n to point 2n (0→0, 1→2, 2→4…). The threads envelope a heart-shaped cardioid — the same multiplication, drawn — as in The Coding Train’s cardioid challenge. Change the multiplier and the whole figure morphs.
Prime factorisation draws too: in a factorisation diagram, each number is arranged from its prime factors, so 6 looks like two groups of three while a prime stays a single unsplittable loop. The structure on screen is the arithmetic.
Common misconceptions
1is not prime. A prime needs exactly two factors;1has only one. This is why factorisations are unique — otherwise you could stick on extra1s forever.2³is not2 × 3. The exponent counts how many times you multiply, so2³ = 2 × 2 × 2 = 8, not6.- Order of operations is not simply left to right. In
2 + 3 × 4you multiply first, giving14; the×binds tighter than the+.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
Work out `47 × 28` using long multiplication. Show the two partial rows.
Answer
`47 × 8 = 376`; `47 × 20 = 940`; `376 + 940 = 1316`. So `47 × 28 = 1316`.
Art hook Draw the answer as an array: a grid `47` columns wide and `28` dots tall, colouring the two partial products in two hues (the `×8` block and the `×20` block) so the split you did on paper becomes two coloured rectangles that stack to the whole.
-
Work out `992 ÷ 32` using long or short division. Is there a remainder?
Answer
`32 × 3 = 96`, so `99 ÷ 32 = 3` remainder `3`; bring down the `2` to make `32`; `32 ÷ 32 = 1`. Answer `31`, with no remainder (check: `32 × 31 = 992`).
Art hook Animate `992` dots being dealt one-by-one into `32` bins arranged in a circle; when every bin holds `31` and none are left over, the ring flashes to show the division came out exact.
-
Write `126` as a product of prime numbers (its prime factorisation).
Answer
`126 = 2 × 63 = 2 × 7 × 9 = 2 × 3 × 3 × 7 = 2 × 3² × 7`. Check: `2 × 9 × 7 = 126`.
Art hook Build a factorisation diagram: `126` shown as 2 big clusters, each split into 3 groups of 3, and each of those into 7 tiny dots — a nested bloom where the shape you see is the prime signature.
-
Find the HCF (highest common factor) and the LCM (lowest common multiple) of `24` and `36`.
Answer
`24 = 2³ × 3`, `36 = 2² × 3²`. HCF = the shared part `= 2² × 3 = 12`. LCM `= 2³ × 3² = 72`.
Art hook Two nested rings of dots, one stepping in 24s and one in 36s around the same circle; mark where a step of each lands on the same point — the first shared landing is the LCM at 72.
-
Work out `20 - 2 × 3² + 4` using the correct order of operations.
Answer
Powers first: `3² = 9`. Then multiply: `2 × 9 = 18`. Then left to right: `20 - 18 + 4 = 6`.
Art hook Show the expression as a tower of coloured tiles that collapse in order — the `3²` tile resolves first, then the `×` tile, then the `+`/`−` tiles — so a viewer watches the rules act like gravity, one layer at a time.
-
What is `√81`? And what is the reciprocal of `5`?
Answer
`√81 = 9` because `9 × 9 = 81`. The reciprocal of `5` is `1/5` (and `5 × 1/5 = 1`).
Art hook Draw a square made of `81` dots (9 by 9); its side length is the square root. Beside it, a pie split into 5 equal slices where one slice is the reciprocal `1/5`.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
List all the factors of `18`.
Answer
`1, 2, 3, 6, 9, 18` — the numbers that divide `18` with no remainder.
Art hook Take 18 dots and show every way to arrange them into a full rectangle: `1×18`, `2×9`, `3×6`. Each rectangle's side lengths are a factor pair, so the factors appear as the shapes that pack perfectly.
-
Is `29` prime or composite? Explain how you know.
Answer
Prime. It has exactly two factors, `1` and `29` — nothing from `2` up to `5` divides it evenly, and `6² = 36 > 29`, so there is no need to test any higher.
Art hook Try to fit `29` dots into any rectangle bigger than a single line — none work except `1×29`. A prime is a number that refuses to form a rectangle; render that stubbornness as a single unbroken loop of dots.
-
Work out `10²`, `10³` and `10⁴`. What do you notice about the zeros?
Answer
`10² = 100`, `10³ = 1000`, `10⁴ = 10000`. The exponent equals the number of zeros.
Art hook A number line drawn on a log-style scale where each power of ten is one equal jump; place a dot at each power so `100`, `1000`, `10000` sit evenly spaced even though the values explode.
-
Work out `36 × 14` using long multiplication. Show the two partial rows.
Answer
`36 × 4 = 144`; `36 × 10 = 360`; `144 + 360 = 504`. So `36 × 14 = 504`.
Art hook A `36`-wide by `14`-tall dot grid, split into a `×4` band and a `×10` band in two colours, stacking to the full 504-dot rectangle.
-
Work out `585 ÷ 15` with short or long division.
Answer
`15 × 3 = 45`, so `58 ÷ 15 = 3` r `13`; bring down the `5` to make `135`; `135 ÷ 15 = 9`. Answer `39` (check: `15 × 39 = 585`).
Art hook Deal 585 dots into 15 equal columns; watch each column climb to height 39 with none left over.
-
Write `2 × 2 × 2 × 2 × 2` using a power. Then find its value.
Answer
`2⁵ = 32`. (Five 2s multiplied together.)
Art hook Start with 1 dot and double it five times: 1→2→4→8→16→32, each doubling shown as a fresh split so the growing cloud of dots visibly counts out `2⁵`.
-
Draw the factor tree for `72` and write its prime factorisation.
Answer
e.g. `72 = 8 × 9 = (2×2×2)×(3×3) = 2³ × 3²`. Check: `8 × 9 = 72`.
Art hook A branching factor tree that grows downward; every leaf is a prime, drawn as a small coloured dot (2s one colour, 3s another), so the finished tree is a two-colour mobile encoding `2³ × 3²`.
-
Spot the mistake: a friend says `4 + 6 × 2 = 20`. What did they do wrong, and what is the right answer?
Answer
They added first. Multiplication comes before addition: `6 × 2 = 12`, then `4 + 12 = 16`. The right answer is `16`.
Art hook Show both readings side by side as two collapsing tile-towers — the wrong one (add first) and the right one (multiply first) — landing on 20 and 16, so the rule's effect is visible as a fork.
-
Find the HCF (highest common factor) of `20` and `30`.
Answer
Factors of `20`: `1,2,4,5,10,20`; of `30`: `1,2,3,5,6,10,15,30`. Shared factors: `1,2,5,10` — the highest is `10`.
Art hook Two overlapping circles (a Venn diagram) of dots; drop each number's factors into the rings and the ones sitting in the overlap are the common factors — the biggest overlap dot is the HCF.
-
Find the LCM (lowest common multiple) of `6` and `8` by listing multiples.
Answer
Multiples of `6`: `6,12,18,24…`; of `8`: `8,16,24…`. The first shared one is `24`.
Art hook Two beads travel around the same track, one hopping 6 steps, one hopping 8; light up the first spot they both land on — 24 — as the least common multiple.
-
Work out `√100 + 2³`. Use order of operations.
Answer
`√100 = 10` and `2³ = 8`; roots and powers first, then add: `10 + 8 = 18`.
Art hook A `10×10` dot square (its side is `√100`) sitting next to a `2×2×2` dot cube; the two shapes' dot-counts add to 18.
-
A number's prime factorisation is `2² × 5`. What is the number, and is it prime or composite?
Answer
`2² × 5 = 4 × 5 = 20`. It is composite, because it is built from more than one prime factor.
Art hook Reverse the factorisation diagram: start from the atoms `2, 2, 5` and assemble them into a single cluster of 20 dots, so the artwork runs the fingerprint backwards into the whole number.