If n is a positive integer, what is the remainder when (7^n + 5) is di

If n is a positive integer, what is the remainder when (7^n + 5) is divided by 10?

(1) When n is divided by 4, the remainder is 1.

\(n=4x+1\)
\(7^(4x+1) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)

(2) When n is divided by 12, the remainder is 5.

\(n=12x+5\)
\(7^(12x+5) + 5\)
unit digit of this number will be \(7+5=12 \)
we get the remainder \(2\) when divided by \(10\)

So D

yashikaaggarwal what is the relation between unit digit cyclicty and finding of remainder ? one thing to know the last digit of number and another thing to know a whole number ... yes cyclycty repeats after 4, so what ?

dammit still have questions with quant

Okay so if we start from start.
we know any no. ending with 1to9 leaves same remainder as its unit digit.
For ex- 11/10 leaves remainder 1. 29/10 leaves remainder 9 and so on.
so we just have to find what's the unit digit of 7^n+5, cool.

now 1st statement says when n is divided by 4 it leaves remainder 1.
then value of n = 1,5,9,13,17............
as stated above 7 unit digit repeats after every 4th power.
and 7^1 unit digit = 7^5 unit digit = 7^9 unit digit (the powers are at the gap of 4)
so the unit digit of every power of 7^5 in this statement is 7
and adding 5 in 7 will give you 12 or simply 2 as unit digit, which will be same when we divide the no. ending with 2 as unit digit with 10.
hence its sufficient.

Similarly for other statement.
Hopefully this helps.

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.