Question

What is the last digit of number 7^2009

Answer 1

@dan815

Answer 2

There is a pattern for 7s

Answer 3

7, 49 , 243 , ... write a couple out see how it begins to repeat

Answer 4

There is more neater ways to do it, but you need to learn some modular arithmetic for it

Answer 5

isn't 7^343..

Answer 6

7^3 = 343*

Answer 7

okay xD

Answer 8

continue on with the pattern

Answer 9

till what point?

Answer 10

tll u see the last numbers repeating

Answer 11

okay did that.. xD

Answer 12

ok so tell me the pattern

Answer 13

7,49,343,2401,16807,117649,823543,5764801,40353607....
so that would be 7,9,3,1,7,9,3,1,7 and so on

Answer 14

very good now lets think, where would 2009 fall in this pattern

Answer 15

Do you have any ideas?

Answer 16

lets see the pattern of the last digits,
**Only*** ~writing last digits~~~~
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
7^6...

Answer 17

I ll let you think about it a little. Tell me what you come up with.

Answer 18

*Hint* every 4 exponents there is a repeating pattern*

Answer 19

think about that 4 and 2009, what can we do about this

Answer 20

i have an idea to sum all the last number until i get 2009 but that'll take some time xD

Answer 21

like 5*7^3+6*7^2+0*7^1 but this doesn't help much..

Answer 22

nope think simpler :)

Answer 23

for example ill give you a hint if i want to find like 2^8 lets say

Answer 24

i know every 4th exponent is 1
because its going
7,9,3,1, 7,9,3,1 7, 9, 3, 1

Answer 25

so like 7^8 = last number is 1
7^12 = last number is 1 again
7^16 = last number is 1

Answer 26

lets think about the closest multiple to 4 around 2009

Answer 27

so that we can work out the last numbers beside that number

Answer 28

for example since i know the pattern is
7 , 9 , 3, 1, 7,9,3,1 7,9,3,1
and if someone asked me what was 2^10 lets say
I would find 2^8 which is 1, then i know
7 , 9 , 3 ,1 7, 9 ,3 ,1
^--- I am there, so 2^10 i have to go up 2 more exponents and i get 9 for the last number.

Answer 29

So in same way for 2009
2008 is a multiple of 4
and 2^2008, last number is 1, so what would 2009 be?

Answer 30

oh my gosh ive been typing 2^2008 and 2^10.. excuse me for that lol it should be 7^2008 and 7^exponents..

Answer 31

it's okay xD

Answer 32

7

Answer 33

yep

Answer 34

Does it make sense?

Answer 35

it does but is there any other way to do this ...'cause this takes a lot of time and won't have much time on the test...

Answer 36

i won't*

Answer 37

hmm

Answer 38

+ i can't use calculator ...

Answer 39

oh really

Answer 40

Have you learnt mods?

Answer 41

i'm familiar with it but no..