What do these numbers have in common?

• Jul 19th 2010, 11:56 AM
wonderboy1953
What do these numbers have in common?
41, 80 and 320?
• Jul 19th 2010, 07:42 PM
Soroban

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

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

• Jul 19th 2010, 07:43 PM
bigblackbronco
they can all be / by 1
• Jul 20th 2010, 07:12 AM
Soroban

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*

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

• Jul 20th 2010, 11:14 AM
wonderboy1953
Keep going.
• Jul 20th 2010, 03:24 PM
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. (Bow)
• Jul 20th 2010, 05:20 PM
wonderboy1953
Quote:

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. (Bow)

That's the answer I was looking for (I wonder how this plays out in binary?)
• Aug 2nd 2010, 09:23 AM
Chokfull
Quote:

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.