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Question

How many non-negative integral values of x satisfy the equation [x5]=[x7]?
(Here [x] denotes the greatest integer less than or equal to x. For example [3.4]=3 and [2.3=3).

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Solution

It's clear that as n increases, n5 will eventually be much larger than n7. As long as the difference is at least 1, the integer parts can't be equal; a (very) little algebra gives us
n5n7+1n7×575=352
There is a corresponding result for n negative, but we don’t have to worry about it.
Of course, it doesn't follow that the integer parts of the quotients are the same for all smaller numbers.
[5×1,7×1)
[5×2,7×2)
[5×3,7×3)
are all obviously no good; dividing by 7 clearly gives a smaller number than dividing by 5. In the middle of this last range we trip over the absolute limit we found before.
Notice that there are five good values of n in [0,5) (not a coincidence), and three good values in the next continuous group of good values, in [7,10) (decreasing from 5 to 3, that is, by 75, again no coincidence). Then there's a lonely individual n in [14,15), and that's it.

So the size of the solution set is 5+3+1=9.

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