Question

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. Please explain and write out all your work! Thank you!!

Answer 1

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.