Question 973731
Guess and check works easily if you can use a computer (preferably with a spreadsheet program),
or a graphing calculator,
but it is time consuming otherwise.
There is an algebra way to get to the solution.
 
You figured out that for the boxes to have dimensions (in centimeters) that are integers, and are in the ratio {{{3:4:5}}} ,
the dimensions (in centimeters) must be {{{3x}}} , {{{4x}}} , and {{{5x}}} ,
with {{{x}}}= a positive integer.
That is one of the requirements of the company Gadgets To Go.
For a box meeting that requirement, the sum of the dimensions (in centimeters) is
{{{3x+4x+5x=12x}}} .
For that box to meet the requirements of the company Ship It Here, it must be
{{{100<=12x<=1000}}} <---> {{{100/12<=x<=1000/12}}} .
Since {{{100/12}}}=8.3333...  and {{{1000/12}}}=83.3333... ,
and {{{x}}} must be integer, it must be {{{9<=x<=83}}} .
We calculate the volume (in cubic centimeters) of a box measuring {{{3x}}} cm by {{{4x}}} cm by {{{5x}}} cm as 
{{{Volume=(3x)*(4x)*(5x)=60x^3}}} .
 
For {{{x=9}}} , we get {{{system(3x=27,4x=36,5x=45,Volume=60*9^3=60*725=43740)}}} .
That agrees with the dimensions you list for the smallest box.
 
For {{{x=83}}} , we get {{{system(3x=249,4x=332,5x=415,Volume=60*9^3=60*725=34307220)}}} .
 
Gadgets To Go seems to have a maximum volume limit lower than 34,307,220 cubic centimeters,
but "The largest they use to ship any product has a volume less then 2 cubic centimeters" does not make sense.
The smallest box Gadgets To Go would ship would measure 3 cm by 4 cm by 5 cm,
and its volume (in cubic centimeters) would be {{{3*4*5=60}}} .
The volume of largest box Gadgets To Go would ship cannot be "less then 2 cubic centimeters."
 
If it were less than {{{V}}} cubic centimeters,
we would require
{{{Volume=60x^3<=V}}} <---> {{{x^3<=V/60}}} <---> {{{x<=root(3,V/60)}}} .
 
If it were that "The largest they use to ship any product has a volume less than 2 cubic meters" = 2,000,000 centimeters,
we would require {{{x<=root(3,2000000/60)}}} , and since {{{root(3,2000000/60)=about32.18}}} , we would settle for integers such that {{{x<=32}}} ,
which meets the {{{9<=x<=83}}} requirement and would result in a largest box with
{{{system(3x=3*32=96,4x=4*32=128,5x=5*32=160,Volume=60*32^3=60*725=1966080)}}} .
That agrees with the dimensions you list for the largest box.