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QUESTION

List the positive divisors of 30, and hence calculate $d(30),
\sigma(30)$ and $\sigma_2(30)$ directly. Now use the formula given
in theorem 8.3 to check your answers.



ANSWER

The positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Thus
$d(30)=$ number of such divisors=8, $\sigma(30)=$ sum of such
divisors=72 and $\sigma_2(30)=$ sum of squares of such divisors=
$1+4+9+25+100+225+900=1300$. From the formula, since 30=2.3.5, we
have $d(30=(1+1)(1+1)(1+1)=8,\
\sigma(30=\frac{2^2-1}{2-1}.\frac{3^2-1}{3-1}.\frac{5^2-1}{5-1}=
\frac{3}{1}.\frac{8}{2}.\frac{24}{4}=72$ and
$\sigma_2(30)=\frac{2^4-1}{2^2-1}.\frac{3^4-1}{3^2-1}.
\frac{5^4-1}{5^2-1}=\frac{15}{3}.\frac{80}{8}.\frac{624}{24}
=5.10.26=1300$.





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