# Thread: What do these numbers have in common?

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

41, 80 and 320?

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

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

3. they can all be / by 1

4. 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*

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

5. Keep going.

6. 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.

7. 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?)

8. 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.