isi-entrance 2015 Q27

isi-entrance · India · UGA 4 marks Number Theory Arithmetic Functions and Multiplicative Number Theory
Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is:
(a) $\frac { k ^ { 2 } } { 72 }$
(b) $\frac { 72 } { k }$
(c) $\frac { 72 } { n }$
(d) none of the above.
(c) Multiply the given sum by $n$.
Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is:\\
(a) $\frac { k ^ { 2 } } { 72 }$\\
(b) $\frac { 72 } { k }$\\
(c) $\frac { 72 } { n }$\\
(d) none of the above.