Factors, multiples, and primes

Activities to try

1. 1

   Factors-and-multiples explorer (three linked modes)

   Why it works: 'Common' literally means the overlap of two sets; completeness of a factor list comes from systematic pairing around the square root; and the first common multiple is the first place two skip-counts collide. Factors are finite and divide into a number (rectangle side-lengths), while multiples are infinite times-table hops.

   1. Factor mode: for a chosen number N (e.g. 36), form every possible rectangle (array) of N counters and log each pair of side-lengths as a factor pair on a factor rainbow — e.g. (1,36), (2,18), (3,12), (4,9), (6,6).
   2. Explain how you know you have found them all: pairs are logged from the outside in, and once you reach the square root the pairs repeat, so the set is complete.
   3. Multiple mode: set one number line to skip-count in 7 and read off the first five multiples (7,14,21,28,35); then set two adjustable number lines to skip-count in 6 and 8, watch where the hops coincide, and record the first THREE common multiples (24, 48, 72) — the first collision is at 24.
   4. Common mode: set two numbers (e.g. 24 and 36), pour each factor list into a two-set Venn diagram, slide the shared factors into the overlap, and identify the largest shared factor (12) as the highest common factor; then repeat with 36 and 48 (shared factors 1,2,3,4,6,12, highest common factor 12).
   5. State in words what 'common' means here: a factor or multiple shared by BOTH numbers — the intersection of the two sets.
2. 2

   Factor detective: prove the list is complete

   Why it works: Systematic searching plus the square-root stopping point guarantees a complete factor list, and 'common' must be justified as membership in both sets.

   1. For a given number, argue aloud or in writing that no factor has been missed: test divisibility in order (1, 2, 3, …) and pair each factor with its partner.
   2. Justify why you can stop checking near the square root: if N = a×b, one of a or b is at most √N, so any larger factor has already appeared as a partner.
   3. Identify what two chosen numbers share and defend your answer: which factors (and which early multiples) belong to BOTH?
3. 3

   Is it prime? Decide and justify the stopping point

   Why it works: Primeness is about the NUMBER of factors, not about being odd; factor-pairing around the square root tells you when it is safe to stop checking.

   1. Decide whether a given number (e.g. 97) is prime.
   2. Test divisibility only by primes up to the square root (for 97: 2, 3, 5, 7, since √97 ≈ 9.8) and record the results.
   3. Explain WHY you can stop at about 10: beyond √97 any factor would pair with one already checked.
   4. Conclude using the definition: a prime has exactly two distinct factors (1 and itself); 1 is neither prime nor composite because it has only one factor.