# Thread: positive integer with 13 factors

1. ## positive integer with 13 factors

What is the lowest positive integer with exactly 13 factors, both the number and 1 count as a factor.

Thanks

2. Hello, reagan3nc!

What is the lowest positive integer with exactly 13 factors?
Both the number and 1 count as factors.

Most numbers have an even number of divisors.

When factoring, the divisors appear in pairs.

. . $\displaystyle 24 \;=\;\begin{Bmatrix}1\cdot24 \\ 2 \cdot 12 \\ 3\cdot 8 \\ 4 \cdot 6 \end{Bmatrix}\;\;\hdots\text{ 8 factors}$

The only numbers with an odd number of factors are squares.

. . $\displaystyle 36 \:=\:\begin{Bmatrix}1\cdot36 \\ 2\cdot 18 \\ 3\cdot12 \\ 4\cdot9 \\ 6\cdot6\end{Bmatrix}\;\;\hdots\text{9 factors}$

The least integer with 13 divisors is: .$\displaystyle 2^{12} \:=\:\boxed{4096}$

. . The divisors are: .$\displaystyle 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096$

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# smallest positive integer with exactly 13 factors

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