Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-008 · ★★☆ Intermediate
What is the least common multiple of \(2^3 \cdot 5\) and \(2 \cdot 3^2\)?
Choose one option.
Show answer
- B. \(360\)
Explanation
Taking the maximum exponent of each prime, the lcm is \(2^3 \cdot 3^2 \cdot 5 = 360\). The option 90 uses the minimum exponent of 2, and the one with 720 adds an extra factor of 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
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- 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-009 How many positive divisors does 60 = 2^2 · 3 · 5 have?
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