Exercise · ZW-PRIME-NUMBER-009 · ★★★ Advanced
If a prime \(p\) divides the product \(ab\), what follows?
Choose one option.
Show answer
- C. \(p\mid a\) or \(p\mid b\)
Explanation
This is Euclid's lemma, valid only for primes. With \(p=6\) (not prime) it fails: \(6\mid3\cdot4\) but 6 divides neither 3 nor 4.
Review the concept: Prime number
A prime number \(p\) is an integer greater than 1 whose only positive divisors are 1 and itself.
\[p>1,\quad d>0,\ d\mid p\ \Rightarrow\ d\in\{1,p\}\]
More practice problems: prime number
- ZW-PRIME-NUMBER-001 Which of these numbers is prime?
- ZW-PRIME-NUMBER-002 How many prime numbers are less than 30?
- ZW-PRIME-NUMBER-003 What is the smallest prime factor of 221?
- ZW-PRIME-NUMBER-004 What is the prime factorization of 360?
- ZW-PRIME-NUMBER-005 To decide whether 97 is prime by trial division, what is the largest prime you must test?
- ZW-PRIME-NUMBER-006 Which of these statements about primes is true?
- ZW-PRIME-NUMBER-007 How many positive divisors does 60 have?
- ZW-PRIME-NUMBER-008 What is the smallest prime greater than 100?