Teaching objectives
A curricular secondary-school lab (ages 12-14), based on the PIM arithmetic notes. All the arithmetic of divisors is told with a single picture: a number is a box of prime bricks (60 = 2 · 2 · 3 · 5). Bricks always have the same colour: 2 brick red, 3 blue, 5 green, 7 yellow and the rest grey.
No calculator: every number handled is at most 200, and its bricks at most 17. Five small factorisations are preferred to one huge one.
What is learned
- Splitting a number down to its primes and discovering that, however you split it, the same bricks come out (the fundamental theorem of arithmetic).
- Recognising primes and composites (1 is neither) and finding them with the sieve of Eratosthenes, which explains why it is enough to cross out up to the prime whose square fits in the table.
- The divisibility tests for 2, 5 and 3 (before the primes, to recognise them better).
- Factorising with the division ladder and writing the result as powers.
- A divisor is a box inside another: m divides n when its bricks are in the box of n, and the ones left over make n/m. Deciding whether a number divides another without dividing.
- Greatest common divisor and least common multiple with a two-circle diagram: the gcd is the bricks in the centre; the lcm, all of them, with the centre ones once. Why being a multiple of 4 and of 6 does not force being one of 24, what coprime numbers are and why gcd · lcm = a · b. Problems about ropes and buses.
- Finding all the divisors of a number: by hand (and seeing that some escape), with the two-prime table, in pairs d · n/d and testing only up to √n.
- Numbers with an odd number of divisors are the perfect squares (even exponents), and the locker problem.
Key mathematical ideas, by block
A · Bricks (rooms 1-5). Primes as unsplittable bricks, uniqueness of factorisation, sieve, tests and the division ladder.
B · Divisors (rooms 6-8). m | n if n = m · k; with bricks, those of m fit in the box of n.
C · Common (rooms 9-13). gcd and lcm with the diagram; lcm(b, c) | a if b | a and c | a; coprimes; gcd · lcm = a · b.
D · All the divisors (rooms 14-20). The table of p^a · q^b (without the general formula, a teaching decision), the smallest number with a given table, pairs, up to √n, odd count ⇔ perfect square, lockers.
Notes for the teacher
- The general formula (α₁ + 1)(α₂ + 1)··· is left out on purpose: divisors are counted with the two-prime table.
- Every room has its explanation in the book on the challenge strip; the discovery rooms (1, 12, 16, 19, 20) show it after the first challenge.
Room-by-room contents
Room 1 · The hammer
Split 120 with the hammer down to its bricks; fundamental theorem in the book after comparing.
Student tasks
- Bet: Ana splits 4 · 30, Luis 6 · 20 — what happens in the end? (the same bricks).
- Split 120 like Ana and like Luis.
- Split 84 freely.
- Build the box of 168 = 2 · 84 without splitting.
Room 2 · Tests for 2, 3 and 5
Tests (before the primes); in the book, animation 234 = 2 · 100 + 3 · 10 + 4 → 2 + 3 + 4 = 9 and a written example (153 and 130).
Student tasks
- Mark multiples of 2, 3 and 5 in three lists.
- Speed test in three rounds with the divisor in big letters: 2 and 5, five numbers (one mistake repeats the round); 3, ten.
Room 3 · The bricks: prime numbers
Primes (exactly two divisors), composites and 1.
Student tasks
- Three «mark the primes» lists with traps (51, 91, 143; the third with 1 and 33).
- Why 1 is not prime.
Room 4 · The sieve of Eratosthenes
Animated sieve on a 50 table (2 crossings per second, 2 s pauses), then 300, up to 17; ends with the primes below 50.
Student tasks
- First new number crossed by 7 (49).
- Next prime after 7 (11).
- Is 11 needed in the 50 table? (no: 121).
Room 5 · Opening the box: factorising
Division ladder of 132 step by step in the book.
Student tasks
- Two prime checks (the second with 1).
- Notebook-style ladder for 84, 90, 126, 231 and 168: each cell is checked on leaving it; starting with a big prime is accepted, with the advice to start with small ones.
Room 6 · Divisors at a glance
What a divisor is: n = m · k.
Student tasks
- 99, 75 and 91: mark the divisors in a list and write the missing one (3, 5, 13).
Room 7 · A box inside another
m | n when the bricks of m are in the box of n; the rest make n/m.
Student tasks
- Take out of 180 a divisor between 40 and 50 (45) and say what is left (4).
- 168: odd divisor above 10 (21; 8 left).
- 150: even divisor between 20 and 40 (30; 5 left).
Room 8 · Does it fit?
Decide without dividing, with both boxes.
Student tasks
- Six yes/no (12 | 180, 8 | 60…).
- Smallest number to multiply 60 by so that 8 divides it (2).
Room 9 · Common bricks
Two-circle diagram with the common bricks in the centre (36 and 60).
Student tasks
- Move the common bricks of 42 and 56, 30 and 45, 48 and 72 to the centre.
Room 10 · Greatest common divisor
gcd = the bricks in the centre.
Student tasks
- gcd of 98 and 140, 120 and 168, 84 and 126.
- Ropes of 90 and 150 cm: longest piece (30) and number of pieces (8).
Room 11 · Least common multiple
lcm = all the bricks, the centre ones once.
Student tasks
- lcm of 14 and 21, 20 and 30, 36 and 54, 24 and 40.
- Buses every 12 and 18 minutes (36).
Room 12 · The 4 and 6 trap
Challenge first, then theory: lcm(b, c) | a; coprimes.
Student tasks
- Multiple of 4 and 6 — of 24? (no: 12); 6 and 7; 6 and 9; 8 and 15.
- A coprime pair (8 and 15).
- Multiple of 10 and 15: surely of 30.
Room 13 · gcd · lcm
gcd · lcm = a · b with the diagram.
Student tasks
- Find the other number: (6, 180, 36) → 30, (4, 120, 24) → 20, (5, 150, 30) → 25.
- lcm from the gcd: 18 and 24, 28 and 42.
Room 14 · All the divisors
Discovery: mark by hand on a grid; only how many are missing or extra is said.
Student tasks
- All the divisors of 84 and of 96.
Room 15 · The table of divisors
Table of 200 = 2³ · 5² (4 × 3), interactive in the book.
Student tasks
- Bet: more divisors, 64 or 60?
- Fill in the tables of 72 and 98.
- Count with an imagined table: 48, 100, 54.
Room 16 · Tables to order
Big exponents on the small primes (theory after the first challenge).
Student tasks
- Smallest number with a table of 6, 9, 8 and 10 cells (12, 36, 24, 48).
Room 17 · Divisors come in pairs
d ↔ n/d with arcs (36 animated in the book).
Student tasks
- Pair up the divisors of 60 and of 100.
- Partner of 12 in 144, 8 in 120, 14 in 196, 15 in 180.
Room 18 · Up to the square root
In each pair one is ≤ √n: testing primes up to √n is enough.
Student tasks
- Up to which prime for 187? (13).
- Are 151, 169, 187, 191 prime?
Room 19 · An odd number of divisors
Discovery: odd count ⇔ perfect square ⇔ even exponents.
Student tasks
- Two «mark those with an odd count» lists.
- Which box is a square; what to multiply 18 by (2).
Room 20 · The lockers
Final mission: 200 lockers and 200 students, animation with ×4 and «Watch again».
Student tasks
- Bet: how many stay open (14).
- How many times locker 36 is touched (9) and why it stays open.
- Lockers touched exactly 3 times (6); last open one (196).
Rooms to project
The most striking ones to show and discuss in class.