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 Find the rule... (Posted on 2003-03-13)
The following are the smallest 9 elements of an infinite set of integers:

0,1,5,6,25,76,376,625,9376

What rule generates the set? What are the next two values?

 See The Solution Submitted by Filias Cupio Rating: 3.4000 (5 votes)

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 Puzzle Solution Comment 7 of 7 |

Each term of the given sequence is an automorphic number. We know that an autiomorphic number is a nonnegative integer T (say) having S digits, such that the last s digits of T^2 when read from left to right constitute the original number, that is T.

For example,

376^2 = 141376, where 376 is has three digits and the last three digits of 376^2 is indeed 376. Hence, 376 is an automorphic number.

We now observe that the two numbers higher than 9376 with this property are 90625 and 109376, since:

90625^2 = 8212890625, and:

109376^2 = 11963109376.

Consequently, the required next two terms of the given sequence are 90625 and 109376.

Edited on May 19, 2008, 5:08 am
 Posted by K Sengupta on 2008-05-19 05:05:17

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