Problem:
The number n can be written in base 14 as abc, can be written in base 15 as acb, and can be written in base 6 as acacc, where a>0. Find the base-10 representation of n.
Solution:
The problem is equivalent to finding a solution to the system of Diophantine equations 196a+14b+c=225a+15c+b and 225a+15c+b= 216a+36c+6a+c, where 1≤a≤5,0≤b≤13, and 0≤c≤5. Simplifying the second equation gives b=22c−3a. Substituting for b in the first equation and simplifying then gives a=4c, so a=4 and c=1, and the base-10 representation of n is 222⋅4+37⋅1=925. It may be verified that b=10≤13.
The problems on this page are the property of the MAA's American Mathematics Competitions