Question

show that 2^55 +1 is divisible by 11

Answer 1

( 25 )11  +  1 

= 3211  +  1 

Now We assume  n  =  11 , SO

= 32n +  1 

And we substitute n  =  1 , And get

= 32  +1  =  33 ( That is divisible by 11 )

Now we substitute n  =  2 , And get

= 322  +1  = 1024  +  1  =  1025 ( That is not divisible by 11 )
And
Now we substitute n  =  3 , And get

= 323  +1  = 32768 +  1  =  32769 ( That is divisible by 11 )
And
Now we substitute n  =  3 , And get

= 324  +1  = 1048576 +  1  =  1048577 ( That is divisible by 11 )

SO, we can say at n  = odd numbers (  1 , 3 , 5 , .... )  we get  32n +  1   is divisible by 11 .

And for our query we have n  =  11 , that is a odd number , SO we can say that 

3211  +  1  is divisible by 11 ,