Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-009 · ★★☆ Intermediate
How many positive divisors does \(60 = 2^2 \cdot 3 \cdot 5\) have?
Choose one option.
Show answer
- A. \(12\)
Explanation
The count is \((2+1)(1+1)(1+1) = 12\). The option 6 does not follow the multiplicative formula, and the one with 8 ignores the factor from the prime 2.
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?
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- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-003 Which integer has the prime factorization 2^3 · 5?
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- ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-005 If a prime p divides the product a · b of two integers, what necessarily follows?
- 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?
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