Exercise · ZW-EUCLID-S-LEMMA-009 · ★★☆ Intermediate
If \(p\) is prime and \(p \mid abc\) with \(p \nmid a\) and \(p \nmid b\), what follows?
Choose one option.
Show answer
- A. \(p \mid c\)
Explanation
Applying the lemma to \((ab)\cdot c\) and using that \(p\) divides neither \(a\) nor \(b\), we get \(p \mid c\). The opposite option contradicts the conclusion.
Review the concept: Euclid's lemma
If a prime \(p\) divides the product \(ab\), then \(p\) divides \(a\) or \(b\). It is the key property that distinguishes primes from composite numbers.
\[p \mid ab \Rightarrow p \mid a \ \lor\ p \mid b\]
More practice problems: euclid's lemma
- ZW-EUCLID-S-LEMMA-001 If p is prime and p | ab, what can you conclude?
- ZW-EUCLID-S-LEMMA-002 Does Euclid's lemma require p to be prime?
- ZW-EUCLID-S-LEMMA-003 Which is the correct statement of Euclid's lemma?
- ZW-EUCLID-S-LEMMA-004 If p is prime and p | 35, which value can p take?
- ZW-EUCLID-S-LEMMA-005 Which statement about a prime p that does not divide a is true?
- ZW-EUCLID-S-LEMMA-006 If 7 | 12x, what can you conclude about x?
- ZW-EUCLID-S-LEMMA-007 If 11 | ab and 11 nmid a, what happens to b?
- ZW-EUCLID-S-LEMMA-008 If p is prime and p | a^2, what follows?