Divisor Sum of Square-Free Integer/Examples/66/Proof 2

Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {66} = 144$

Proof

We have that:

$66 = 2 \times 3 \times 11$

Hence:

\(\ds \map {\sigma_1} {66}\)

\(=\)

\(\ds \paren {2 + 1} \paren {3 + 1} \paren {11 + 1}\)

Divisor Sum of Square-Free Integer

\(\ds \)

\(=\)

\(\ds 3 \times 4 \times 12\)

\(\ds \)

\(=\)

\(\ds 3 \times 2^2 \times \paren {2^2 \times 3}\)

\(\ds \)

\(=\)

\(\ds 2^4 \times 3^2\)

\(\ds \)

\(=\)

\(\ds \paren {2^2 \times 3}^2\)

\(\ds \)

\(=\)

\(\ds 12^2\)

\(\ds \)

\(=\)

\(\ds 144\)

$\blacksquare$