He has between 10 and 20 batteries. He counts the batteries by putting them into piles of 4 and finds that he has 3 left over. He then counted them by putting them into piles of 3 and found that he had one left over. How many batteries does he have?

15 batteries

Step-by-step explanation:

If he puts the batteries in piles of 4, and there is 3 left over, the number of batteries he can have is:

2 piles: 4*2 + 3 = 11

3 piles: 4*3 + 3 = 15

4 piles: 4*4 + 3 = 19

Then, when he puts in piles of 3, there is no batteries left over, so the number of batteries is a multiple of 3.

From the 3 possibilities above (11, 15 and 19), only the number 15 is a multiple of 3, so the number of batteries is 15.