Proofs Unveiled – The “End Digit”

Have you ever written an algebraic proof at school or college? Proofs have become such a widespread aspect of modern algebra and geometry that I can’t already imagine today’s curriculum without them… Nevertheless, many students regard proofs as a torture. They really are, if you don’t know how to deal with them.

To solve a proof, you need to remember one thing – proofs can be grouped in categories. There are geometric proofs and algebraic proofs, and both of them can be subdivided into more groups. Today I’ll talk about one kind of proof – the end digit proof – which is my favorite!

Here’s an example: what is the end digit of the expression 222222? Such an example is herculean unless you apply logic to it. And it’s not difficult! In fact, it’s almost as easy as you can even imagine! To solve such a problem, here’s what we’d think:

1. In the original base (the number to the left of the exponent), only the end digit matters. After all, when you multiply two numbers, only the final digit contributes to the end digit of the product. So, we need to leave off everything but the very last digit of the base. In other words, you’ve mentally simplified the problem to  2222, which is easier to think through, since you need to keep track of one digit.
2. Find the correlation between the exponent and the end digit of the answer. Let’s use small exponents, starting from 1. Then let’s try to find a relationship between the exponent and the end digit. If we do find one, then we can trace the relationship all the way to the needed exponent without physically examining every single exponent. Here’s how we’d do it:

2¹  = 2

2²  = 4

2³  = 8

2⁴  = 16

2⁵  = 32

Aha, we’ve found a cycle! It repeats itself every 4 exponents. Since at the beginning the end digit is 2 and it is the same at the end of the cycle, we can conclude that the pattern repeats every 4 steps. The sequence of end digits (starting from the exponent 1) is 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, … The pattern keeps going on indefinitely, so we have no need to look further. We’ve found the relationship!

Now we just need to trace this pattern forward to our needed exponent – 222.

Let’s start with the exponent of 1, which correlates to an end digit of 2. Let’s add 200 to the exponent. Since 200 is a multiple of 4 (4 is the length of the cycle), the position in the cycle doesn’t change, so the exponent 201 also correlates to an end digit of 2.

Let’s add 20 to the exponent. 20 is also a multiple of 4, so an exponent of 221 still matches an exponent of 2.

Now, if we try to add more multiples of 4, we’ll exceed our initial exponent of 222. So let’s instead follow the cycle.

Since 222 is one more than 221, we’ll be one step down the cycle with an end digit of 4. So this is our answer.

To summarize, we need to:

1. Shrink the initial base to the very last digit, since only it affects the end digit of the answer,
2. Find a correlation between the exponent and the end digit, by working with small exponents (usually start with 1),
3. Find the cycle’s length,
4. Work your way to the needed exponent by repeatedly adding multiples of the cycle’s length, whenever possible.
5. Then, when you’re really close, just follow the exponent-end digit correlation until you’re done!

Hope my post helped you in some way! 🙂