1985 AHSME Problems/Problem 29
Problem
In their base representations, the integer consists of a sequence of eights and the integer consists of a sequence of fives. What is the sum of the digits of the base representation of ?
Solution
Notice that by the formula for a geometric series.
Similarly, .
Thus, .
We can multiply out to get .
We now find this in decimal form. , where there is one and zeroes.
, where there is two and zeroes.
We subtract to find that , where there are nines, eight, and zeroes.
We now add to get , where there are nines, eight, zeroes, one, and a final zero.
Next, we begin to divide by . We get this to be , where there are ones, zero, eights, nine, and a final zero.
Finally, we have to multiply by . Doing this, we find that the pattern continues, and the final outcome is , where there are fours, three, fives, six, and a final zero. Adding this up, the sum of the digits is .
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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