# decimal digit as final digit

• February 25th 2008, 12:28 PM
mandy123
decimal digit as final digit
Which decimal digits occur as the final digit of a forth power of an integer?
• February 25th 2008, 05:58 PM
ThePerfectHacker
Reduce $x^{4}$ mod $10$. You can just reduce $0^4,1^4,...,9^4$ by mod $10$ since it will repeat. And those are the possibilities.
• February 25th 2008, 07:18 PM
topsquark
Quote:

Originally Posted by mandy123
Which decimal digits occur as the final digit of a forth power of an integer?

Think of a decimal number as
$a_n \cdot 10^n + ~...~ + a_1 \cdot 10 + a_0$

So when we raise this to the 4th power we get something like
$b_{4n} \cdot 10^{4n} + ~...~ + b_1 \cdot 10 + a_0^4$

So the last digit will be given by the last digit of the original number raised to the fourth power.

The possibilities are
$1 \implies 1$

$2 \implies 6$ (16)

$3 \implies 1$ (81)

$4 \implies 6$ (256)

etc.

I get {1, 5, 6}.

-Dan