+934 Let's break the sum of two numbers down to see if we can find a pattern:
\(17^1+15^1=32\) The highest power of 2 that can divide this is 32 or \(2^5\)
\(17^2+15^2=514\) The highest power of 2 that can divide this is 2 or \(2^1 \)
\(17^3+15^3=8288\) The highest power of 2 that can divide this is 32 or \(2^5 \)
\(17^4+15^4=134146\) The highest power of 2 that can divide this is 2 or \(2^1\)
Here you can see a pattern, the largest power of 2 that can divide the sum is either \(2^5\) or \(2^1\). You can also see that when the exponents in the sum are odd, the sum is divisible by \(2^5\) but when the exponents of the sum are even, the sum is divisible by \(2^1\).
In the sum of the numbers above, their exponents are 2 apart.
\(17^3+15^1=4928\) The highest power of 2 that can divide this is 64 or \(2^6\)
\(17^4+15^2=83746\) The highest power of 2 that can divide this is 2 or \(2^1\)
\(17^5+15^3=1423232\) The highest power of 2 that can divide this is 128 or \(2^7\)
\(17^6+15^4=24188194\) The highest power 2 that can divide this is 2 or \(2^1\)
\(17^7+15^5=411098048\) The highest power of 2 that can divide this is 64 or \(2^6\)
This pattern is different, but we can see that every other odd exponent in the sum has \(2^6\) as the largest power of 2 divisor.
Based on this pattern, \({17}^{17}+{15}^{15}\), the largest divisor that is a power of 2 is \(2^7\).
+934 Sorry for the late response and long answer...I made an error in my explanation and had to correct it!
+934 I guess my solution is wrong...can you elaborate a little more on yours please?
Sorry, don't really have any fancy explanation to put forward. Just wrote this short computer code and it came up with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 as powers of 2 that divide the expression:
n=1; a=if((15^15 + 17^17) % 2^n==0, goto2, goto3);printn,", ",;n++;if(n<100, goto1, 0)
