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 the integer ?
Solution
By the formula for the sum of a geometric series, and similarly so
We now compute the decimal expansion of this expression. Firstly, , with one and zeroes, and , with two and zeroes. Subtracting therefore gives where there are nines followed by eight and then zeroes. Adding transforms this to , now with nines followed by eight, zeroes, one, and a final zero.
Using long division, and noting that and , it follows that with ones, zero, then eights, nine, and a final zero. Lastly, using long multiplication and noting that , , and , we obtain where there are fours, three, fives, six, and a final zero, so 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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