The Last Digit of 5 to the Power of n

The Last Digit of 5 to the Power of n

What happens when we multiply a number by itself over and over again? In mathematics, this is called raising a number to a power. Today, we are going to explore a beautiful pattern that emerges when we look at the powers of the number five.

Let's write out the first few powers of five. Five to the first power is just five. Five squared is twenty-five. Five cubed is one hundred and twenty-five. And five to the fourth power is six hundred and twenty-five. Look closely at the last digit of each result. It is always five!

Why does this happen? Think about the mechanics of multiplication. Whenever you multiply any number ending in five by another five, the final step of the multiplication is always five times five, which equals twenty-five. This guarantees that the final digit of the answer will always be five, no matter how many times we repeat the process.

Let's look closely at what happens when we multiply by five. Every time we increase the power, we are just taking the previous result and multiplying it by another five. Let's write out this chain visually.

Imagine we have some number ending in five. When we multiply it by five, we focus entirely on that last column, the units place. Five times five is twenty-five. The two carries over, but the unit digit is guaranteed to be a five.

Because the units place of the product is solely determined by multiplying the units digits of the factors, this cycle locks in. A units digit of five multiplied by five will always produce a units digit of five, over and over, forever.

To prove this pattern holds forever, let's use a simple algebraic representation. Any whole number that ends in the digit five can be written as ten times some integer, k, plus five.

Now, let's see what happens when we multiply this number, N, by five. We distribute the five to both terms, which gives us fifty k plus twenty-five.

We can rewrite twenty-five as twenty plus five. Factoring out a ten from the first two terms reveals a multiple of ten plus five. Since ten times any integer ends in zero, adding five guarantees the final digit is always five!

We have explored the pattern, visualized the multiplication, and proved it rigorously using mathematical induction. Now, we can confidently state our final verdict for every single natural number n.

No matter how unimaginably large n becomes—whether it is five, five million, or a googolplex—the last digit of five to the power of n is locked in. It is always, beautifully, five.

This is the magic of mathematical proof. With a simple, two-step ladder of induction, we tamed infinity. We don't need to calculate five to the power of a billion to know its last digit; we know it with absolute certainty.