Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-007 · ★★☆ Intermediate
If \(n = 2^a \cdot 3^b\) and \(n = 72\), what are \(a\) and \(b\)?
Choose one option.
Show answer
- C. \(a = 3, b = 2\)
Explanation
\(72 = 8 \cdot 9 = 2^3 \cdot 3^2\), so \(a = 3\) and \(b = 2\). With \(a = 2\) one gets 36, and with \(b = 3\) one gets 216.
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-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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