Babbage’s Number

October 9, 2018

Charles Babbage, whose Analytical Engine was a direct predecessor of today’s digital computer, gave this example of a problem that his Analytical Engine could solve in an 1837 letter to Lord Bowden:

What is the smallest positive integer whose square ends in the digits 269,696?

Babbage knew that 99,736 has a square with the required ending, but didn’t know if there was a smaller number.

Your task is to find Babbages’s number. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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