The Unit's Digit Trivia Quiz

Answer: 5

An even integer must end in one of the following digits: 0, 2, 4, 6, 8. So no even integer can end in 5.

Answer: 6

Note that we are only interested in the last digit of the sum. Adding up the unit's digits of the three numbers gives 2 + 6 + 8 = 16, hence the unit's digit of this sum is 6.

Answer: 3

To find the unit's digit of the product, just multiply the unit's digit of these two numbers to get 9 * 7 = 63. So the unit's digit of the product is 3.

Answer: 1

If we divide 2008 by 4, we get 502 with a remainder of 0. So the unit's digit of 2^2008 is the same as the unit's digit of 3^0 = 1.

Answer: It is always equal to 6

Observe that 4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024, ...

The unit's digits of the powers of 4 alternate 4, 6, 4, 6, 4, ...
So we see that if we raise 4 to an even power, the unit's digit will always be 6.

Answer: 7

Let's see what happens when we raise 7 to powers: 7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5 = 16807, ... So the unit's digits of powers of 7 follow a cycle of length 4: 7, 9, 3, 1, 7, ...

So let's find the remainder when we divide 101 by 4:

101 = 25*4 + 1

Since the remainder is 1, 7^101 has the same unit's digit as 7^1, which is 7.

Answer: 1

First of all, 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, ... so the unit's digits of powers of 3 go in the sequence 3, 9, 7, 1, 3, ...

4^1 = 4, 4^2 = 16, 4^3 = 4, so the unit's digits of powers of 4 alternate 4, 6, 4, ...

Divide 15 by 4 first: 15 = 3*4 + 3. Since the remainder is 3, the unit's digit of 3^15 is the same as the unit's digit of 3^3, which is 7. Now divide 15 by 2: 15 = 7*2 + 1. Since the remainder is 1, the unit's digit of 4^15 is the same as that of 4^1, or 4. Thus the unit's digit of 3^15 + 4^15 is obtained by adding the numbers 7 + 4 = 11. Therefore the unit's digit of the answer is 1.

Answer: 8

First do the addition in the parentheses. When you add 1234 and 5678, the answer will have units digit 2 (4 + 8 = 12). When you cube the result, the answer will have unit's digit 2^3 = 8.

Answer: It is always 1

You can check that if you raise 5 to any positive integer power, the unit's digit of the answer will always be 5. Likewise, if you raise 6 to any positive integer power, the unit's digit will always be 6. Therefore, when you add the results, you'll be getting 5 + 6 = 11, so 1 is the unit's digit of the answer.

Answer: 5

If a positve integer that is greater than 10 ends in 5, then that number is divisible by 5 and hence cannot be prime. However, all the other answers are fine since 11, 13, and 19 are all prime.

One thing I should mention: The arithmetic I described in the above ten questions is really arithmetic "modulo 10". We say that two integers a and b are congruent mod 10 if a - b is divisible by 10. In other words, positive integers a and b are congruent mod 10 if and only if they have the same unit's digit. For much more on "modular arithmetic", check out any book on the subject of "Number Theory". I hope you enjoyed the quiz! Thanks for playing!