A prime that divides a square
6 × 6 = 36. Every prime that divides 36 divides 6.
Step 1 of 4
- 6 breaks into 2 × 3. Squared: 36.
- 3 divides 36. And 3 divides 6. True.
- 4 also divides 36, since 36 ÷ 4 = 9.
- But 4 does not divide 6. 4 is not prime.
The rule
A lemma is a small fact that bigger proofs use. This one is used all through the chapter.
Take a whole number n above 0. If a prime divides n², it divides n.
Why? Squaring doubles every power in n's list of primes. So n² has no new primes. Each number has one set of primes. So n² has exactly the primes of n. A prime that divides n² is one of those primes. So it divides n.
Where it fails
4 divides 36 = 6², but 4 does not divide 6. The lemma is about primes. 4 = 2 × 2 is not prime.
9 fails too. 9 divides 36, but not 6.
Some non-primes, like 6, divide 36 and also divide 6. But the lemma says nothing about non-primes.
Now try it. Pick k. Look for a leak.
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