Problem:
A rational number written in base eight is ab.cd, where all digits are nonzero. The same number in base twelve is bb.ba. Find the base-ten number abc.
Solution:
The integer parts of the two representations must match, so abeight=bbtwelve. This implies 8a+b=12b+b, from which a=23b. Because both a and b must be positive integers less than 8, the only two possibilities for the ordered pair (b,a) are (2,3) and (4,6). For b=4 and a=6 the fractional part of the number equals 0.46twelve =124+1446=83=0.30eight , so d would be 0. On the other hand if b=2 and a=3, then the fractional part is 0.23twelve =122+1443=163=0.14eight , and c=1 and d=4. Indeed, 32.14eight =22.23twelve . The requested number is 321.
The problems on this page are the property of the MAA's American Mathematics Competitions