Question 203146
One would hope that there would be a better way than actually multiplying 14 3's and then subtracting 1. Well there is.<br>
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 <b>any</b> 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 <b>no</b> effect on the ones digit of the product.<br>
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.<br>
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":
{{{3^14 = ((3^3)*(3^4))^2 = (27*81)^2}}}
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!