Question

If $a,b,c,d,....$ are positive integers, whose sum is equal to $n$, show that the least value of
$|\underset{–}{a}|\underset{–}{b}|\underset{–}{c}|\underset{–}{d}......$ is ${\left(|\underset{–}{q}\right)}^{p-r}{\left(|\underset{–}{q+1}\right)}^r$,
where $q$ is the quotient and $r$ the remainder when $n$ is divided by $p$

Answer 1

Given $n=pq+r$

Therfore sum $n$ is not exactly divisible by $p$.

For given $p$ numbers , $m$ are equal to $q$ and remaining $p-m$ are each equal to $q+1$.

Product of the factorials of unequal integer whose sum is constant is minimum, when they are equal or when they are consecutive that is they differ by unity.

The sum of the quantities $=n$

$mq+(p-m)(q+1)=n$

$mq+pq+p-mq-m=n$

$pq+p-m=n$ or $pq+r=n$

$p-m=r$

$m=p-r$

Least value of \underline$|a\underset{–}{|b}\underset{–}{|c}\underset{–}{|d}...=(\underset{–}{|q}\underset{–}{|q}\underset{–}{|q}...)m$ times $(\underset{–}{|q+1}\underset{–}{|q+1}...)p-m$ times

Least value of \underline$|a\underset{–}{|b}\underset{–}{|c}\underset{–}{|d}...={\underset{–}{|q}}^m{\underset{–}{|q+1}}^{p-m}$

Least value of \underline$|a\underset{–}{|b}\underset{–}{|c}\underset{–}{|d}...={\underset{–}{|q}}^{p-r}{\underset{–}{|q+1}}^r$