Question 543190
The key to solving this problem is to realize that your mystery number is 2 short of a multiple of 84, 72, and 96. If we call your mystery number {{{n}}}, then {{{n+2}}} is a multiple of 84, 72, and 96.
{{{84=7*12=2^2*3*7}}}
{{{72=8*9=2^3*3^2}}}
{{{96=32*3=2^5*3^2}}}
The least common multiple of 84, 72, and 96 is
{{{LCM(84,72,96)=2^5*3^3*7=96*7=672}}}
If the problem just asked for any "number which when divided by 84, 72 and 96 leaves remainder 82, 70, and 94 respectively," we would say that {{{n+2=672}}}, with {{{n=670}}} would be one of many solutions. All the solutions would be numbers such that
{{{n+2=672*k}}} for some positive integer {{{k}}}.
As the problem asks for "the largest 5-digit number," we need to look for an {{{n+2}}} multiple of 672 that makes {{{n}}} the largest 5-digit number complying with the condition.
{{{10^5/672=about148.8}}} according to my calculator, so I can easily estimate that {{{n+2=148*672}}} will yield a 5-digit {{{n}}}, while the next larger multiple of 672 {{{n+2=149*672}}} will yield a 6-digit {{{n}}}
In other words, {{{n=148*672-2=99454}}} is the solution.