First, understand this is a fairly difficult problem for a six year old. Most adults will be unable to do it also.
Start him off as thinking about it in terms of sets or collections of objects.
This is all the numbers that are greater than 12 but less than 32
That is all the possible answers. Obviously some of them are wrong but we have narrowed it down to 19 different numbers of guys in the band.
For them to play in groups of three the number has to be divisible by 3. 5 guys in the band cannot play in groups of 3. Why? Take away the first group of 3 and you have 2 left over that is not a group of 3. Now 6 guys can play as 2 groups of 3. We go through all the possibilites and pick out only the numbers of guys that also form groups of 3. We get
Look at that group above. Each number, say 18 guys can play in groups of 3. Prove that that by actually making the groups of 3. You will see that it forms 6 perfect groups of 3. So we now only have 7 possiblities (the ones above)
For them to play in groups of four the number has to be divisible by 4. 18 guys in the band cannot play in groups of 4. Why? Take away the first group of 4 and you have 14 left over
Take away the second group of 4 and you 10 left. Take away the next group of 4 and you have 6. Now take away the next and you have 2 left over. Now 8 guys can play as 2 groups of 4. We go through all the possibilites and pick out only the numbers of guys that also form groups of 4. We get
We are down to one number. One last test too. Can 24 form groups of 6? Yes it can 6, 6 , 6 , 6. 4 perfect groups of 6. You are done, 24 is the answer.