Starting with 0, a mathematician labels every non-negative integer as one of five types: alpha, beta, gamma, delta, or

The pattern of labeling numbers is a repeating pattern of 5 letters.

So we need to determine how many cycles plus any fraction of a cycle are contained within the number.

We are free to choose which gamma and which delta. Let's choose the smallest, 2 and 3.

So, the number for which a label is desired is:

2^7+3^7

We want to map this to the appropriate label, so we can start with determining the number of cycles of 5:

(2^7+3^7)/5

Nobody wants to calculate this, so let's look at remainders of 5, which is determined by the last digit of each number:

2^1 =2. 3^1=3
2^2=4. 3^2=9
2^3=8. 3^3= 7
2^4=6 3^4=1

These last digits then repeat in a pattern of 4. So the numbers are 1 cycle of 4 plus 3 steps into the new cycle, or 8 and 7.

8+7 = 15

15 leaves no remainder when divided by 5, so there are a whole number of cycles of 5 in the number.

Since beta is assigned to 1 and alpha to 5 completes the cycle, the appropriate label is alpha, A