Find all values of positive integer n for which a number 2^(2n)+1 is divisible by 5

Answer 1

Hi Emma,

I know this was posted a while ago, but to answer this question for you, and in case others would like to have reference, we can utilize the power rule of exponents to start off and make the problem a little simpler.

I can rewrite 2²ⁿ + 1 as (2²)ⁿ + 1 because when we raise an exponent to an exponent, it's the same as multiplying the two exponents. Now that I have undistributed the exponents, we can take 2² first which is 4.

So now, I have 4ⁿ + 1. If we take a look at the pattern this establishes for every value of n starting with n = 1:

4¹ + 1 = 4 + 1 = 5, which is divisible by 5.

4² + 1 = 16 + 1 = 17, which is not divisible by 5.

4³ + 1 = 64 + 1 = 65, which is divisible by 5.

4⁴ + 1 = 256 + 1 = 257, which is not divisible by 5.

4⁵ + 1 = 1024 + 1 = 1025, which is divisible by 5.

4⁶ + 1 = 4096 + 1 = 4097, which is not divisible by 5.

Notice how every time you multiply 4 by another 4, the value will alternate ending in a 4 or 6, so when I add 1 to the value I will get a number that ends in 5 or 7.

Of course, every number that ends in 5 is divisible by 5, so that means every other exponent starting from 1 will simplify the expression to a number that is divisible by 5.

Therefore, a way to express all positive values of n that would make this expression divisible by 5 is n = 2k + 1, where k can be any whole number greater than or equal to zero (+1 because you want to start with 1).

Hope this helps!

Thanks!