What are the last three digits of the number 7^411? How would I go about solving this? Thanks!

Guest Feb 27, 2015

#1**+10 **

Note that 7^11 ends in 743

And 7^100 ends in 001

And 7^200 ends in 001 because we would be multiplying (7^100) x (7^100) which would have the following result in the last 3 digits......

........001

x ........001

-----------------

..........001

........ 00

.........0

----------------

.............001

So.....7^400 would also end in 001 because we would be squaring 7^200 = (.......001) x (........001) which would have the same three ending digits as above

Therefore 7^411 = 7^200 x 7^200 x 7^11 would end in

.................001

.................743

------------------

.........003

.........04

.........7

--------------------

.................743

And 743 is the answer.....!!!

CPhill
Feb 27, 2015

#1**+10 **

Best Answer

Note that 7^11 ends in 743

And 7^100 ends in 001

And 7^200 ends in 001 because we would be multiplying (7^100) x (7^100) which would have the following result in the last 3 digits......

........001

x ........001

-----------------

..........001

........ 00

.........0

----------------

.............001

So.....7^400 would also end in 001 because we would be squaring 7^200 = (.......001) x (........001) which would have the same three ending digits as above

Therefore 7^411 = 7^200 x 7^200 x 7^11 would end in

.................001

.................743

------------------

.........003

.........04

.........7

--------------------

.................743

And 743 is the answer.....!!!

CPhill
Feb 27, 2015