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Math Help - What do these numbers have in common?

  1. #1
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    What do these numbers have in common?

    41, 80 and 320?
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  2. #2
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    Um . . . Their binary representations begin with "101" ?


    . . \begin{array}{ccc} 41 &=& 101001_2 \\<br />
80 &=& 1010000_2 \\<br />
320 &=& 101000000_2 \end{array}

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  3. #3
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    they can all be / by 1
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    Don't know how I missed that . . .


    Quote Originally Posted by bigblackbronco
    they can all be / by 1

    And I wasted all that time looking for patterns . . . *sigh*


    . . \begin{array}{ccc}<br />
41 &=& 4^2 + 5^2 \\<br />
80 &=& 4^2 + 8^2 \\<br />
320 &=& 8^2 + 16^2 \end{array}

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  5. #5
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    Keep going.
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  6. #6
    Senior Member eumyang's Avatar
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    If you add the three numbers you get a perfect square:
    41 + 80 + 320 = 441 = 21^2

    And... if you add any two of the numbers you also get perfect squares:
    \begin{aligned}<br />
41 + 80 &= 121 = 11^2 \\<br />
80 + 320 &= 400 = 20^2 \\<br />
41 + 320 &= 361 = 19^2<br />
\end{aligned}

    If it wasn't for Soroban's last post I wouldn't have thought of this.
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  7. #7
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    Quote Originally Posted by eumyang View Post
    If you add the three numbers you get a perfect square:
    41 + 80 + 320 = 441 = 21^2

    And... if you add any two of the numbers you also get perfect squares:
    \begin{aligned}<br />
41 + 80 &= 121 = 11^2 \\<br />
80 + 320 &= 400 = 20^2 \\<br />
41 + 320 &= 361 = 19^2<br />
\end{aligned}

    If it wasn't for Soroban's last post I wouldn't have thought of this.
    That's the answer I was looking for (I wonder how this plays out in binary?)
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  8. #8
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    Quote Originally Posted by wonderboy1953 View Post
    That's the answer I was looking for (I wonder how this plays out in binary?)
    It shouldn't change a bit in binary as this solution has nothing to do with the numbers' digits, only their values.
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