What do these numbers have in common?

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)

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


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

 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)

Don't know how I missed that . . .


bigblackbronco said:
they can all be / by 1

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


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

 
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Jan 2010
278
138
If you add the three numbers you get a perfect square:
\(\displaystyle 41 + 80 + 320 = 441 = 21^2\)

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

If it wasn't for Soroban's last post I wouldn't have thought of this. (Bow)
 
Oct 2009
769
87
If you add the three numbers you get a perfect square:
\(\displaystyle 41 + 80 + 320 = 441 = 21^2\)

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

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?)
 
May 2009
108
4
Neverland
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.