Problem Solving

What is the remainder when 3^50 is divided by 8?

is it 3?

i'll explain if yes. thanks.

The answer is a 1 IMO

here is how.
The last digit of powers of 3 follow this pattern : 1,3, 9, 7. (ie for 3^0, 3^1, 3^2 and 3^3)

So now we have to figure out the units digit of 3^50 and then divide it by 8 to find the remainder.

units digit of 3^50 is 9. How. 50/4 will give u remainder of 2.
ie units digit of 3^50 is same as units digit of 3^2.

Looking up the list above we get units digit of 3^2 as 9.

9 divided by 8 leaves a remainder = 1.

HT helps

IMO 1


3- divided by 8 remainder 3
9- divided by 8 remainder 1
27- divided by 8 remainder 3
81- divided by 8 remainder 1
.
.
.
so for all the odd powers you get remainder 3
and for all even powers you get remainder 1
answer is 1

The answer is 1.

since 3^50 can be rewritten as 9^25.

So any multiples of 9 divided by 8 will get you the reminder of 1.

So ans is 1.

Regards
suresh

vittalgmat wrote:
units digit of 3^50 is 9. How. 50/4 will give u remainder of 2.
ie units digit of 3^50 is same as units digit of 3^2.

Hello Vittalgmat,
can you please explain how the above conclusion is made. Is there a generic theory available?
regards

missionGMAT007 wrote:

vittalgmat wrote:
units digit of 3^50 is 9. How. 50/4 will give u remainder of 2.
ie units digit of 3^50 is same as units digit of 3^2.

Hello Vittalgmat,
can you please explain how the above conclusion is made. Is there a generic theory available?
regards

Yes there is general theory..The powers of numbers (the common ones) follow a repeated pattern.( i mean the units digits).
in this case, the units digits for the powers of 3 follow this pattern
1, 3, 9, 7.
Try out some powers.. and u will see ..( u can use a calculator to prove this point quickly!! )

So units digit of 3^0 = units digit of 3^4 = .. so on.
similary units digit of 3^1 = unit digit of 3^5 .. so on..


here are some patterns for other numbers
It is worth memorizing .. or atleast be aware..

Quick list of repeatable units digit for 2-12 powers:


2 = 2, 4, 8, 6
3 = 3, 9, 7, 1
4 = 4, 6
5 = 5
6 = 6
7 = 7, 9, 3, 1
8 = 8, 4, 2, 6
9 = 9, 1
10 = 0
11 = 1
12 = 2, 4,8, 6