How many of the first one thousand perfect fourth powers have either 1 or 6 as their units digit? the answer is 800 fourth powers. explain and write out all your work!

Let's see.

Let's look at the first 10 natural numbers raised to the 4th power.

1^4 = 1 * 1 * 1 * 1 = 1

2^4 = 2 * 2 * 2 * 2 = 16

3^4 = 3 * 3 * 3 * 3 = 81

4^4 = 4 * 4 * 4 * 4 = 256

5^4 = 5 * 5 * 5 * 5 = 625

6^4 = 6 * 6 * 6 * 6 = 1296

7^4 = 7 * 7 * 7 * 7 = 2401

8^4 = 8 * 8 * 8 * 8 = 4096

9^4 = 9 * 9 * 9 * 9 = 6561

10^4 = 10 * 10 * 10 * 10 = 10000

Notice that of the first 10 natural numbers raised to the 4th power, only 5^4 and 10^4 do not have a 1 or a 6 as its units digit. All other 8 numbers have a 1 or a 6 as their units digit.

Now think of 11. If you raise 11 to the 4th power, it will end in 1 just 1^4 does. 12^4 will end in 6 just like 2^4 does. This holds true for the 4th powers of all of the natural numbers. Every natural number that ends in 1 has a 4th power that ends in 1. Every natural number that ends in 2 has a 4th power that ends in 6, etc. Since 8 out of 10 4th powers have a 1 or a 6 in the units digit, then out of 1000, you get 8/10 * 1000 = 800.