Question 1041463
{{{N}}}= the number we are looking for.
Since {{{N}}} divided by {{{26}}} leaves the remainder {{{6}}} ,
{{{N=26q+6}}} where {{{q}}} is a non-negative integer that is the quotient of {{{N}}} divided by {{{26}}} .
{{{N=26q+6}}} means {{{N+20=26q+6+20=26q+26=26(q+1)}}} ,
so {{{N+20}}} is a multiple of {{{26}}} .
If {{{N}}} divided by {{{36}}} yields the quotient {{{p}}} and leaves the remainder {{{16}}} ,
{{{N=36p+16}}} , and {{{N+20=36p+16+20=36p+36=36(p+1)}}} is a multiple of {{{36))) .
If {{{N}}} divided by {{{56}}} yields the quotient {{{m}}} and leaves the remainder {{{36}}} ,
{{{N=56m+36}}}, and {{{N+2=56m+36+20=56m+56=56(m+1)}}} is a multiple of {{{56}}} .
So, {{{N+20}}} is a common multiple of {{{26}}} , {{{36}}} , and {{{56}}} .
The least positive {{{N+20}}} is the least common multiple of
{{{26-2*13}}} , {{{36=4*9}}} and {{{56=4*14=2*4*7}}} , which is
{{{2*4*7*9*13=56*9*13=6552}}} ,
so the least {{{N}}} is {{{N=6552-20=highlight(6532)}}} .