What do these numbers have in common?

**wonderboy1953** · July 19th 2010, 10:56 AM

41, 80 and 320?

**Soroban** · July 19th 2010, 06:42 PM

Um . . . Their binary representations begin with "101" ?

. . $\begin{array}{ccc} 41 &=& 101001_2 \\ 80 &=& 1010000_2 \\ 320 &=& 101000000_2 \end{array}$

**bigblackbronco** · July 19th 2010, 06:43 PM

they can all be / by 1

**Soroban** · July 20th 2010, 06:12 AM

Don't know how I missed that . . .

Originally Posted by bigblackbronco

they can all be / by 1

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

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

**wonderboy1953** · July 20th 2010, 10:14 AM

Keep going.

**eumyang** · July 20th 2010, 02:24 PM

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:
$\begin{aligned} 41 + 80 &= 121 = 11^2 \\ 80 + 320 &= 400 = 20^2 \\ 41 + 320 &= 361 = 19^2 \end{aligned}$

If it wasn't for Soroban's last post I wouldn't have thought of this.

**wonderboy1953** · July 20th 2010, 04:20 PM

Originally Posted by eumyang

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:
$\begin{aligned} 41 + 80 &= 121 = 11^2 \\ 80 + 320 &= 400 = 20^2 \\ 41 + 320 &= 361 = 19^2 \end{aligned}$

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?)

**Chokfull** · August 2nd 2010, 08:23 AM

Originally Posted by wonderboy1953

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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