13. How many infinite arithmetic sequences of positive integers are there which contain the numbers 3 and 39 ?
Answer: 12
Solution: The common difference should be a positive divisor of $39-3=36=2^{2} \cdot 3^{2}$, which has $(2+1)(2+1)$ factors. If the common difference is 1 , then any one of 1,2 , or 3 may be the first term. If the common difference is 2 , then the first term may be either 1 or 3 only. For the other 7 possible common differences, the first term must be 3 . Thus, there are 12 such arithmetic progressions in all.