Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-010 · ★★☆ Intermediate
Let \(n = 900\). How many positive divisors does \(n\) have?
Choose one option.
Show answer
- D. \(27\)
Explanation
\(900 = 2^2 \cdot 3^2 \cdot 5^2\), so the number of divisors is \((2+1)(2+1)(2+1) = 27\). The option 9 omits one of the three primes, and 18 does not come from multiplying the factors \((2+1)\).
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-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?