What is the remainder when 3^123 is divided by 5?

hi,
another approach after the highlighted portion
So need to look at the \(\ -2^{123}\)
\(\ -2^{123}= (-2^4)^{\frac{123}{4}} = 16^{30}*-2^3=(15+1)^{30}*-8\)
now\((15+1)^{30}\) will leave 1 as remainder, so total remainder = 1*-8 = -8..
as correctly done, we cannot have negative remainder so remainder = 10-8 = 2

Simple and straight to the point!

Q. 3^123 % 5

= (5-2)^123 % 5
* all the factors will be divisible by 5 except (-2)
= (-2)^123%5
* taking in account the cyclicity of 2; ie 4
= (-2^4)^30*(-2)^3 % 5
= 16^30*(-16) % 5
* 16%5 leaves remainder as 1
= (1)^30*(-16) % 5
= (-16) % 5 leaves remainder as -1; ie 5 + (-1) = 4

We know to find what is the remainder when \(3^{243}\) is divided by 5

Theory: Remainder of a number by 5 is same as the unit's digit of the number

(Watch this Video to Learn How to find Remainders of Numbers by 5)

Using Above theory , Let's find the unit's digit of \(3^{243}\) first.

We can do this by finding the pattern / cycle of unit's digit of power of 3 and then generalizing it.

Unit's digit of \(3^1\) = 3
Unit's digit of \(3^2\) = 9
Unit's digit of \(3^3\) = 7
Unit's digit of \(3^4\) = 1
Unit's digit of \(3^5\) = 3

So, unit's digit of power of 3 repeats after every \(4^{th}\) number.
=> We need to divided 243 by 4 and check what is the remainder
=> 243 divided by 4 gives 3 remainder

=> \(3^{243}\) will have the same unit's digit as \(3^3\) = 7
=> Unit's digits of \(3^{243}\) = 7

But remainder of \(3^{243}\) by 5 cannot be more than 5
=> Remainder = Remainder of 7 by 5 = 2

So, Answer will be C
Hope it helps!

Watch the following video to learn the Basics of Remainders

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.