Problem:
Suppose n is a positive integer and d is a single digit in base 10. Find n if
810n=0.d25 d25 d25…
Solution:
Since 810n=0.d25 d25 d25…, we have 1000810n=d25.d25 d25…. Subtracting gives
810999n=1000810n−810n=d25=100 d+25.
Consequently 999n=810(100d+25), which leads to 37n=750(4d+1). Noting that 750 and 37 are relatively prime, we see that 4d+1 must be a multiple of 37. Since d is a single digit, d=9 and hence n=750.
The problems on this page are the property of the MAA's American Mathematics Competitions