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Prime Factorization and Divisors

Learn how to break down numbers big and small, as proposed by the Fundamental Theorem of Arithmetic. See more

Counting Divisors

How many positive divisors does 20 have?

Which of these numbers has the most divisors?

Which of these numbers has the most divisors?

\[ \begin{array} {c|c|c} n & \text{Prime Factorization} & \text{# of Divisors} \\ \hline 20 & 2^2 \times 5^1 & 6 \\ 30 & 2^1 \times 3^1 \times 5^1 & 8 \\ 50 & 2^1 \times 5^2 & 6 \\ 60 & 2^2 \times 3^1 \times 5^1 & 12 \\ 90 & 2^1 \times 3^2 \times 5^1 & 12 \\ 150 & 2^1 \times 3^1 \times 5^2 & 12 \\ 210 & 2^1 \times 3^1 \times 5^1 \times 7^1 & ? \end{array} \] How many divisors does 210 have?

\[ \begin{align} 210 &= 2^1 \times 3^1 \times 5^1 \times 7^1\\ \\ 5 &= 5^1\\ 15 &= 3^1 \times 5^1\\ 28 &= 2^2 \times 7^1\\ 35 &= 5^1 \times 7^1\\ 42 &= 2^1 \times 3^1 \times 7^1 \end{align} \] Which is not a divisor of 210?

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