Sigma and Tau functions

Number Theory Level 3

The sum of divisors function denoted by $\sigma(n)$ is the sum of all positive divisors of $n$ , and
the number of divisors function denoted by $\tau(n)$ is the number of positive divisors of $n$ .

If a number $n$ is divisible by 2 and 3 and that $\sigma(n)=363$ and $\tau(n)=10$ , find $n$ .

The answer is 162.

1 solution

Renz Mina
Jul 31, 2016

We have $n=p_1^{a_1}p_2^{a_2}$ since the factors of $10$ . But since $2$ and $3$ are factors, then $n=2^{a_1}3^{a_2}$ . Also, we have that $(\frac{2^{a_1+1}-1}{2-1})(\frac{3^{a_2+1}-1}{3-1})=363$ The possible values of $a_1+1$ and $a_2+1$ are $(5,2)$ , $(2,5)$ , $(1,10)$ and $(10,1)$ . Hence $a_1+1=2$ and $a_2+1=5$ . $a_1=1$ and $a_2=4$ $n=162$

Sigma and Tau functions

The answer is 162.

1 solution

0 pending reports