let

x = the number of 16 bulb boxes

y = the number of 24 bulb boxes

the bulbs from x + the bulbs from y = total number of bulbs

16*x + 24*y = 232 divide both sides by 8

2*x + 3*y = 29

since y >=0 we have 2*x = 29 - 3*y <=29 that is x<=29/2

x < = 14

also notice that x is not divisible by 3 (otherwise if x is divisible by 3 the left side of the equation 2*x + 3*y = 29 will be divisible by 3 while the right side 29 is not divisible by 3)

If x = 1 then y = (29 - 2)/3 = 9 is one solution x=1, y=9

If x = 2 then y = (29 - 4)/3 is not integer thus is not a solution

If x = 4 then y = (29 - 8)/3 = 7 is second solution x=4, y=7

We found 2 possible ways of packing:

One way: 1 box of 16 bulbs & 9 boxes of 24 bulbs.

Second way: 4 boxes of 16 bulbs & 7 boxes of 24 bulbs.