If N denotes the set of all positive integers | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The function $f(n)$ represents the sum of all positive divisors of $n$. For a number $n = p_1^{a_1} \cdot p_2^{a_2} \cdots$,the sum of divisors is

Vedclass · Accepted Answer

$4(2^{k+1}-1)$

Vedclass · Answer

(D) The function $f(n)$ represents the sum of all positive divisors of $n$. For a number $n = p_1^{a_1} \cdot p_2^{a_2} \cdots$,the sum of divisors is given by $\sigma(n) = \frac{p_1^{a_1+1}-1}{p_1-1} \cdot \frac{p_2^{a_2+1}-1}{p_2-1} \cdots$. Given $n = 2^k \cdot 3^1$,the sum of divisors is: $f(2^k \cdot 3) = (1 + 2 + 2^2 + \cdots + 2^k) \cdot (1 + 3)$. Using the geometric series sum formula $\sum_{i=0}^{k} 2^i = \frac{2^{k+1}-1}{2-1} = 2^{k+1}-1$. Thus,$f(2^k \cdot 3) = (2^{k+1}-1) \cdot (4) = 4(2^{k+1}-1)$.

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) =$ the sum of positive divisors of $n$,then $f(2^k \cdot 3)$,where $k$ is a positive integer,is

Similar Questions

If $P(A \cup B)=0.8$ and $P(A \cap B)=0.3$,then $P(\bar{A})+P(\bar{B})$ is equal to:

Radius of gyration of a body depends on

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin to the plane, then the equation of the plane is

Which one of the following molecules contains both ionic and covalent bonds?

Vedclass Test Series

Exam Paper Generator

Online Exam Module