Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-005 · ★☆☆ Basic
If a prime \(p\) divides the product \(a \cdot b\) of two integers, what necessarily follows?
Choose one option.
Show answer
- A. \(p \mid a\) or \(p \mid b\)
Explanation
Euclid's lemma says that if a prime divides a product, it divides at least one of the factors. The other options reverse the divisibility or require two conditions at once, which the lemma does not claim.
Review the concept: Fundamental theorem of arithmetic
Every integer greater than 1 can be written uniquely as a product of primes, up to the order of the factors. It is the foundation of most of elementary number theory.
\[n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}\]
More practice problems: fundamental theorem of arithmetic
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-001 What is the prime factorization of 12?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-002 Two factorizations of 30 are 2 · 3 · 5 and 5 · 3 · 2. What does the fundamental theorem of arithmet…
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-003 Which integer has the prime factorization 2^3 · 5?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-004 How many distinct prime factors does 42 have?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-006 What is the prime factorization of 360?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-007 If n = 2^a · 3^b and n = 72, what are a and b?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-008 What is the least common multiple of 2^3 · 5 and 2 · 3^2?
- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-009 How many positive divisors does 60 = 2^2 · 3 · 5 have?
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