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?
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
x^2+x-222y=0
A quadratic in x whose determinant is 888y+1
The only single digit for which this is a perfect square is y=6.
888*6+1=5329=73^2
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.)
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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
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Posted by Jer
on 2019-04-18 09:38:52 |
Two Solutions
