Prime numbers

Activities to try

1. 1

   Rectangle-building: when is only 1×n possible?

   Why it works: A prime number can only be made into the 1×n rectangle (exactly two factors); a composite makes at least one other rectangle; and 1 is neither prime nor composite because it has only a single factor.

   1. Take a number and try to arrange that many counters (or squares on grid paper) into complete rectangles with more than one row and more than one column (a single line, 1×n, does not count as another rectangle).
   2. For 51, find that a 3×17 rectangle can be built, so 51 has factors other than 1 and itself — it is composite.
   3. For 97, try rows of 2, 3, 5 and 7 — the primes up to its square root (just under 10), since any larger factor would pair with a smaller one already tested — and find that only a single 1×97 line forms, so 97 is prime.
   4. Try the number 1: it forms only a single 1×1 square — one factor — so it fits neither the prime nor the composite group.
   5. Record each verdict with its factor pair as justification (e.g. '51 = 3×17, composite'; '97, only 1×97, prime').
2. 2

   Sieve of Eratosthenes on a hundred square

   Why it works: Composite numbers are exactly the multiples that get crossed out; once 1 is set aside as neither, the survivors are the primes. This also makes the primes to 19 visible for recall.

   1. On a 1–100 grid, first cross out 1 (it is neither prime nor composite), then circle 2 and cross out every later multiple of 2.
   2. Move to the next uncrossed number (3) and cross out its multiples; repeat for 5, then 7.
   3. Notice that the numbers still standing (circled or uncrossed) are exactly the primes, and that every number crossed out as a multiple is a multiple of a smaller prime, so it is composite (1 was crossed out separately because it is neither).
   4. Highlight the primes up to 19 (2, 3, 5, 7, 11, 13, 17, 19) for memorisation.
3. 3

   Efficient primality testing to the square root

   Why it works: For a whole number greater than 1, you only need to check divisibility by primes up to the square root; if none divide the number, it is prime. Divisibility rules make this fast.

   1. To test whether a whole number greater than 1 is prime, divide only by the primes 2, 3, 5, 7… up to its square root (1 is neither prime nor composite, so it is not tested this way).
   2. Use divisibility shortcuts: even → ÷2; digit sum divisible by 3 → ÷3; ends in 0 or 5 → ÷5.
   3. Test 97: not even, digit sum 16 (not divisible by 3), does not end in 0/5, not divisible by 7 — and since the next prime 11 has 11×11 = 121 > 97, there is no need to test further, so 97 is prime.
   4. Test 51: digit sum 5+1 = 6, divisible by 3, so 51 = 3×17 → composite.