Last digits

Hello, and welcome to Myriora Maths! What is the last digit of seven to the power of one hundred? That number has eighty-five digits! But we can find its last digit in seconds.

First, what does to the power of mean? Seven to the power of three means three sevens, multiplied together. Seven, times seven, times seven: three hundred and forty-three. So seven to the power of one hundred is one hundred sevens, multiplied together!

Let's look at the last digits. Seven to the power of one is seven. Seven squared is forty-nine: the last digit is nine. Seven cubed is three hundred and forty-three: the last digit is three. Seven to the power of four is two thousand, four hundred and one: the last digit is one. Then it starts again: seven, nine, three, one.

Why does it repeat? When we multiply, only the last digit decides the next last digit. Nine times seven is sixty-three, so after nine comes three. Three times seven is twenty-one, so after three comes one. And one times seven brings us back to seven.

So the pattern repeats every four. One hundred divided by four is twenty-five, remainder zero. Remainder zero means we land at the end of a cycle: the fourth digit. So the last digit of seven to the power of one hundred is one!

A common mistake is reading the remainder the wrong way. Remainder one means the first digit in the cycle. But remainder zero means the last one, not the first! Seven to the power of one hundred ends in one, not seven.

Your turn! What is the last digit of three to the power of fifty? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! The last digits of three go three, nine, seven, one. Fifty divided by four is twelve, remainder two. The second digit in the cycle is nine!

Quick review. Write out the last digits until they repeat. Divide the power by the length of the cycle. The remainder tells you where you land, and remainder zero means the end of the cycle. See you next time!