Divisor Sum of Square-Free Integer/Examples/70/Proof 2
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Example of Divisor Sum of Square-Free Integer
- $\map {\sigma_1} {70} = 144$
Proof
We have that:
- $70 = 2 \times 5 \times 7$
Hence:
| \(\ds \map {\sigma_1} {70}\) | \(=\) | \(\ds \paren {2 + 1} \paren {5 + 1} \paren {7 + 1}\) | Divisor Sum of Square-Free Integer | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times 6 \times 8\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times \paren {3 \times 2} \times 2^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3^2 \times 2^4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2^2 \times 3}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 12^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 144\) |
$\blacksquare$