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Simple Adding (Posted on 2019-04-18) Difficulty: 2 of 5
I have a positive integer X.

When I add all the positive integers from 1 to X, I get a result of YYY, where Y is a positive integer from 0 to 9.

What is X?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Two Solutions Comment 1 of 1
I already knew 666 is a triangular number, but are there other 3-repdigit triangular numbers?

Using the variables from the problem
The sum of 1 to x is (x^2 + x)/2
Represent the result as 111y
Equating the two and rearranging we get


A quadratic in x whose determinant is 888y+1

The only single digit for which this is a perfect square is y=6.  
This leads back to x=36.

(It doesn't fit, but the next value of y is 25, which leads to x=74 and the not-actually-a-reptile 2775.  You can see the 2500+250+25 though.)


A simpler way to look at the problem:

111y=3*37y so either x or x+1 has 37 as a factor.

x=36 gives 666
x=37 gives 703

  Posted by Jer on 2019-04-18 09:38:52
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