Euclid 2020/Problem 4

(a) The positive integers and have no common divisor larger than . If the difference between b and a is and, what is the value of ?

(b) A geometric sequence has  first term and common ratio . An arithmetic sequence has  first term and common difference . The ratio of the term in the geometric sequence to the term in the geometric sequence equals the ratio of the term in the arithmetic sequence to the term in the arithmetic sequence. Determine all possible values of . (An arithmetic sequence is a sequence in which each term after the  first is obtained from the previous term by adding a constant, called the common difference. For example, are the  first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the  first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, is a geometric sequence with three terms.)