Squares repeat the beads

The idea

Theorem 1.2: if a prime p divides a², then p divides a, where a is a positive integer; this follows from the Fundamental Theorem of Arithmetic.

Beads of 30

30 = 2 × 3 × 5. So 30² = 900 has the same prime beads, each one twice.

Could some prime divide 900 without dividing 30?

• Yes, some prime might
• No, never
• Not sure

Hunt for a prime

Tap the primes. Does any prime divide a² without dividing a?

Now try numbers that are not prime. Take a = 6, so a² = 36.

Theorem 1.2

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Why? Write a = p₁ p₂ … pₙ (primes). Then a² = p₁² p₂² … pₙ²: the same primes, each twice. By the uniqueness in the Fundamental Theorem of Arithmetic, a² has no other prime. So a prime that divides a² is one of p₁, p₂, …, pₙ, and it divides a.

The word prime matters: 4 divides 6² = 36, but 4 does not divide 6.

Notes

If a prime p divides a², then p divides a (a a positive integer). It follows from the Fundamental Theorem of Arithmetic.

Check yourself

The prime 5 divides a². What must be true?

12 divides 36 = 6². Does 12 have to divide 6?

a² = 2² × 5² × 11². How many different primes divide a?

Answer: 3

2, 5 and 11 divide a². By Theorem 1.2 each divides a. So three primes divide a.

• 5 divides a — correct. Yes! By Theorem 1.2, a prime that divides a² divides a.
• 25 divides a. Not promised. Take a = 5: a² = 25, and 25 does not divide 5.
• a = 5. a could be 10, 15, 20 … Any multiple of 5 works. Only “5 divides a” is promised.
• 5 may not divide a. Theorem 1.2 says it always does, because 5 is prime.
• Yes, by Theorem 1.2. Theorem 1.2 is only for a prime p. 12 is not prime.
• No: 12 divides 36 but not 6 — correct. Yes! 12 = 2 × 2 × 3 needs two 2s. 36 has them (one from each 6) but 6 has just one.
• Yes, because 6 × 6 = 36. That does not follow. 12 is bigger than 6, so it cannot divide 6.