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

Answer 1

for 2^(2n+1) no integer n will ever make it divisible by 5

for 2^(2n) + 1, if n is divisible by 5, or any odd integer, the expression will be divisible by 5, except n=0,

n=5,10,15, 20, 25, ...500, 505, ... 23645,23650,...

n= 1,3,5, 7, 9, 11, ....501,503,....23647, 23649,...

combine them, take the union of the sets

n= 1,3,5,7,9,10,11,13,15,17,19,20, 21, 23,25, .... are all integers n that make 2^(2n)+1 divisible by 5

n=0 2^0 +1 = 1+1= 2

n=1, 2^2+1 =4+1= 5 5/5=1

n=3, 2^6+1 = 64+1 =65, 65/5 =13

n=5 2^10+1 = 1024+1 = 1025, 1025/5 = 205

n=7 2^14+1 = 19385, 19385/5 = 3877

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