What is the smallest positive integer that when divided by , and leaves the remainders , and , respectively?

Source: NCTM Mathematics Teacher, August 2006

**Solution**

If leaves a remainder of , the possible values of are . The integers form an arithmetic progression with first term = and common difference = . The value of the th term of is

Similarly, they form arithmetic progressions with first terms = and common differences = . The values of their respective th terms are given below

Note that the are multiples of . The smallest such multiple is the LCM of which is . Thus, the smallest positive integer that when divided by leaves a remainder of respectively is .

**Answer**: