Is 2^n + 3^n divisible by 5?

Question

Is  2^n + 3^n  divisible by 5?

(1)  n = (2p)^2 
(2) p/2 is an odd integer

Self made. Suggestions welcome.

chetan2u · Answer

hi nalin,

a good try, where you have tried to test the number properties..

However statement II may not fit in...
a) firstly p in this case will be infinity.
b) in Actuals you may not find such language..

Intention of statement II is correct but it could be given through 
a) p/2 is an odd integer
or b) p+ k > 8 where k <2 and p and k are integers...

nalinnair · Answer

Edited as suggested.

mvictor · Answer

1.
if we have n an even number, then 2^n + 3^n definitely is not divisible by 5
if we have n an odd number, then 2^n + 3^n definitely is divisible by 5.

we are told that n = (2p)^2 -> n=4* p^2
if p is an integer, then we have the first option...never divisible
if p is 0 - then we have 2^0 + 3^0 = 2. not working
if p is a fraction, suppose 1/2 -> p^2 = 1/4 -> n=1, then the sum is divisible.

so, we need more information on p.

2. p/2 is an odd integer. -> p is an even integer. but alone, definitely not sufficient.

1+2.
we know for sure that p is an even integer, in which case 2^n + 3^n - is never divisible by 5.

answer is C.

attached - powers of 2 and 3 up to n=19.
we can see the pattern here that when n is odd, then the sum of the two is divisible by 5.

p.s. "(2) There are p stars in the milky way galaxy." - what an original way of saying that p is an integer :D

Is 2^n + 3^n divisible by 5?

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