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The penultimate digit (Posted on 2018-11-12) Difficulty: 2 of 5
A problem from the 1996 Georg Mohr mathematics competition in Denmark:

n is a positive integer. The next-to-last digit in the decimal expression
of n^2 is 7.
What’s the last digit?

Please provide your reasoning, not only the digit.

No Solution Yet Submitted by Ady TZIDON    
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soln Comment 2 of 2 |

n = {…xy}   where {} means concatenates digits 


n^2 = {…7m}


where m is one of (0,…9)  


sup. y^2 = {b1 b2}

        xy   = {c1 c2}


set up the n^2 multiplication in the grade school way:


    {     …x       y}

 X {    …x       y}

_______________ notes:

 (c2)                    [carries for x (xy)]

 (c1)  (b1)           [carries for y (xy)]

         c2      b2    [y(xy)]

+       c2              [x(xy)]

___________________________

       7       m


{… 7} = b1 + 2 c2, so b1 must be odd


the squares from 1 digit integers that have an odd ten’s place are (16 and 36) = y^2 = {b1 b2}

b2 is the ones place of y^2, so, b2 = m = 6. n^2 ends in 76


Edited on November 12, 2018, 5:48 pm
  Posted by Steven Lord on 2018-11-12 17:46:50

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