Two factorizations of 30 are 2 · 3 · 5 and 5 · 3 · 2. What does the fundamental theorem of arithmetic say about them?

They are the same factorization, up to the order of the factors

Only the one that lists the primes in increasing order is valid

Neither is valid, because 30 has no prime factorization

They are different, because the order of the factors changes

Exercise · ZW-FUNDAMENTAL-THEOREM-OF-ARITHMETIC-002 · ★☆☆ Basic

Two factorizations of \(30\) are \(2 \cdot 3 \cdot 5\) and \(5 \cdot 3 \cdot 2\). What does the fundamental theorem of arithmetic say about them?

Choose one option.

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D. They are the same factorization, up to the order of the factors

Explanation

The theorem guarantees that prime factorization is unique up to the order of the factors. Since \(2 \cdot 3 \cdot 5\) and \(5 \cdot 3 \cdot 2\) have the same primes with the same exponents, they describe the same decomposition.

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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.

\[n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}\]

Two factorizations of \(30\) are \(2 \cdot 3 \cdot 5\) and \(5 \cdot 3 \cdot 2\). What does the fundamental theorem of arithmetic say about them?

Explanation

Review the concept: Fundamental theorem of arithmetic

More practice problems: fundamental theorem of arithmetic