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

lets 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 thus one solution is 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 thus one solution is x=4, y=7

If x = 5 then y = (29 - 10)/3 is not integer thus is not a solution

If x = 7 then y = (29 - 14)/3 = 5 thus one solution is x=7, y=5

If x = 8 then y = (29 - 16)/3 is not integer thus is not a solution

If x = 10 then y = (29 - 20)/3 = 3 thus one solution is x=10, y=3

If x = 11 then y = (29 - 22)/3 is not integer thus is not a solution

If x = 13 then y = (29 - 26)/3 = 1 thus one solution is x=13, y=1

If x = 14 then y = (29 - 28)/3 is not integer thus is not a solution

232 bulbs can be packed in to full boxes in the following 5 ways:

1 box of 16 bulbs & 9 boxes of 24 bulbs

4 boxes of 16 bulbs & 7 boxes of 24 bulbs

7 boxes of 16 bulbs & 5 boxes of 24 bulbs

10 boxes of 16 bulbs & 3 boxes of 24 bulbs

13 boxes of 16 bulbs & 1 box of 24 bulbs