SOLUTION: find all the positive integers 'x' which when expressed in decimal notation have both of the folloing properties a)the firs (left-most) digit is 6 b) when this digit is removed,

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Question 430331: find all the positive integers 'x' which when expressed in decimal notation have both of the folloing properties
a)the firs (left-most) digit is 6
b) when this digit is removed, the remaining integer is x/25
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose our n-digit integer is x+=+6%2A10%5E%28n-1%29+%2B+sum%28a%5Bi%5D10%5Ei%2C+i+=+0%2C+n-2%29. Then,



(multiply both sides by 25)

24%2Asum%28a%5Bi%5D10%5Ei%2C+i+=+0%2C+n-2%29+=+6%2A10%5E%28n-1%29

sum%28a%5Bi%5D10%5Ei%2C+i+=+0%2C+n-2%29+=+%2810%5E%28n-1%29%29%2F4+=+25%2A10%5E%28n-3%29

This simply looks for the (n-1)-digit numbers whose decimal representation is 25000...0. This means that the "remaining" number can be 25, 250, 2500, ... We put the 6 in front, so all integers 625, 6250, 62500, ... satisfy.