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Problem #1 (squares and ending digits) watch

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    Find the sum of all two digit numbers, whose squares end in the digits "01".

    Please make yuour answers coloured in white so that you don't spoil it for other people trying to have a go.
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    2 digit not including "01" "02" ... "09"?
    Please make yuour answers coloured in white so that you don't spoil it for other people trying to have a go.
    unless you're using TSR black and alien blue colourschemes
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    yes, a two digit number x meaning 9<x<100.
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    199.

    working coming up....
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    le the digits be ab

    1 <= a <= 9
    0 <= b <= 9

    (10a+b)² = 100a² + 20ab + b²

    last digit depends only on b²

    b = 1 => b² = 1
    b = 9 => b² = 81

    for b = 1, 20ab has to end in 0

    => 2a has to end in 0

    => a = 5, hence 51 is one of the numbers

    for b = 9, 20ab has to end in 0

    => 8 + 18a has to end in 0

    => a = 4 or 9. hence 49 and 99 are solutions

    => sum is 49+51+99 = 199
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    a² = 1 mod 100
    So a² = 1 mod 10 (I chose 10 because it's a nice balance of checking squares mod 10 and checking which of those work mod 100)
    odd squares end in 1, 9, 5, 9, 1
    So a² = 1 or 9 mod 10
    A quick check shows that:
    99 + 51 + 49 = 199
 
 
 
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Updated: January 11, 2005

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