2007 iTest Problems/Problem 45
Revision as of 05:05, 16 June 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 45 — grinding the base numbers)
Problem
Find the sum of all positive integers such that , where represent distinct base digits, .
Solution
Using the definition of base numbers, the equation can be rewritten as
To find the values of , use casework for values of since has the most influence on the value of . Casework will be heavy, but a few tips can lighten the load. First, since is one less than a multiple of , is congruent to or modulo . Second, once , can not be higher for a given . Third, use estimation to approximate the lower bound for a given .
- If , then is just more than . The first few values of that work are , , , and . Testing each case, when and .
- If , then is just more than . The first few values of that work are , , , and . Testing each case, when and .
- If , then is just more than . The first few values of that work are , , . Testing each case, when and .
- If , then is just more than . The first few values of that work are and . After testing each case, no values of work when .
- If , then is just more than . The first few values of that work are and . After testing each case, once again, no values of work when .
- If , then is just more than . The first few values of that work are and . After testing each case, yet again, no values of work when .
In summary, the only possible values of are and , and the sum of the values equals .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 44 |
Followed by: Problem 46 | |
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