find the last 4 digits of 9 to the 2095 power

[nowayitsmel]

i need help determining the last four digits of:

9 to the 2095 power?

 

[stapel]

...the last four digits of 9 to the 2095 power

What powers have you found so far? (You've done 9[sup:1hlj9ham]0[/sup:1hlj9ham], 9[sup:1hlj9ham]1[/sup:1hlj9ham], 9[sup:1hlj9ham]2[/sup:1hlj9ham], etc; but how far did you get?)

What pattern have you noticed?

Please be complete. Thank you!

Eliz.

 

[nowayitsmel]

i saw a pattern in the last digit of 1 and 9 and since the number 2095 ends wth an odd number the last digit will be 9.
i also saw the pattern 0 8 2 6 4 4 6 2 8 0 for the second to last number and i saw the pattern and saw that this was the pattern for 10 times. so i divided 2095 by 10 and got 209.5 and then i just counted 5 from the starting point and ended up with 4 being the second to last number.

but now the third digits are:
0 0 7 5 0 4 9 7 4 4 but i can't figure out the pattern?

 

[stapel]

but now the third digits are:
0 0 7 5 0 4 9 7 4 4 but i can't figure out the pattern?

Just as the pattern in the 10's place was longer (having a period of nine) than the pattern in the 1's place (having a period of two), so also the pattern for the 100's place will be longer, and the pattern for the 1000's place will be longer still. :shock:

Note that, since you only care about the last four digits, you can work with just those digits. When you go up to the next power, all you're really doing is multiplying the last number by 9. When you multiply the last four digits by 9, you will then get the last four digits of whatever would have been the next power. In this way, you can keep using your calculator. :wink:

Keep going, until you find a pattern. It may take some time, but you will get there! Have fun!

Eliz.

There is a way to do this using the Binomial Theorem, but if you've never heard of it, it would probably take longer to study it, than to just plug-n-chug into your calculator.