Exercise · ZW-EUCLID-S-LEMMA-003 · ★☆☆ Basic
Which is the correct statement of Euclid's lemma?
Choose one option.
Show answer
- C. If \(p \mid ab\), then \(p \mid a\) or \(p \mid b\), with \(p\) prime
Explanation
The correct statement requires \(p\) to be prime and concludes divisibility of one of the factors. The other options reverse or alter the hypotheses.
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-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?
- ZW-EUCLID-S-LEMMA-009 If p is prime and p | abc with p nmid a and p nmid b, what follows?