Singapore Mathematical Olympiad (SMO)

Question

Two distinct 2-digit prime numbers p, q can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of p and q, find all possible pairs p, q.

Accepted Answer

The average of p, q is p+q2. Therefore 2(100p + q) = (p+q)k for some integer k. Thus 200p + 2q = (p+q)k 198p = (p+q)(k-2). Since p, q are primes, (p+q) 198. Now 11 + 13 = 24 p+q 89 + 97 = 186. Since 198 = 2 3^2 11, the only even factor of 198 that lies between 24 and 186 is 66. Thus p+q = 66 and there are 4 pairs: (13, 53), (19, 47), (23, 43), (29, 37).