SOLUTION: What is the last digit of this number: 3^14

SOLUTION: What is the last digit of this number: 3^14 - 1 ?

Question 203146: What is the last digit of this number: 3^14 - 1 ?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
One would hope that there would be a better way than actually multiplying 14 3's and then subtracting 1. Well there is.

The simple way is based on the the idea that one can just work the the last digits of products. The key is this: Whenever you multiply any two integers, the ones digit of the product is determined solely by the ones digits of the numbers being multiplied. The other digits, whatever they may be, will have no effect on the ones digit of the product.

Think about it. Try multiplying some integers. Notice how the ones digit of the product comes from the ones digit of the product of the two ones digits. For example: The ones digit of the product of 24,487 * 100,349 will be the ones digit of the product of 7 and 9. 7*9 = 63 so the ones digit of 24,487 * 100,349 will be a 3. All the other digits will not change the fact that the ones digit will be 3. The ones digit of the product of 949 * 30877 will also be a 3 for the same reason.

With this idea in mind then we can concern ourselves solely with the ones digit of 3^14. And we can figure this out by breaking 3^14 into easily calculated "parts":

and then calculating just the ones digits. The ones digit of the product of 27 and 81 will be 7. And if we square a number that ends in 7 we will get a number that ends in 9. So 3^14 ends with a 9. So 3^14 - 1 will end in an eight!
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