For <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> a positive integer, let <img alt="$f(n)$" height="18" src="https://latex.artofproblemsolving.com/e/e/f/eef0203e65664bd00b5f0325c5c5fcd81bcead8d.png" width="34" /> be the quotient obtained when the sum of all positive divisors of <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> is divided by <img alt="$n.$" height="8" src="https://latex.artofproblemsolving.com/b/d/9/bd923e41635af40a1abe58f230732f2280b0723a.png" width="14" /> For example, <img alt="\[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\]" height="37" src="https://latex.artofproblemsolving.com/a/3/0/a30498646d780aea6d03d5130640089c4775112f.png" width="278" /> What is <img alt="$f(768)-f(384)?$" height="20" src="https://latex.artofproblemsolving.com/0/5/9/059ad8410129b7b6d6472694ce2e970886214e31.png" width="137" />

Question No: 356

For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example,

$f(14)=(1+2+7+14)\div 14=\frac{12}{7}$

What is $f(768)-f(384)?$

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Tags: Algebra

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