For a positive integer x, what is the remainder when 3^x+1 is divided

==> If you modify the original condition and the question, for the remainder question when divided by 10, the exponent has the period of 4. In other words, you get \(~3^1=~3, ~3^2=~9, ~3^3=~7, ~3^4=~1\), so the ones digits has the period of 3-->9-->7-->1-->3-->…. Thus, for con 1), from x=4n+2, the exponent always gets a period of 4, and since you get \(3^4^n^+^2+1=(~3^2)+1=(~9)+1=~0\), the remainder is always 0, hence it is sufficient.

Therefore, the answer is A.
Answer: A

Sorry for pulling you back to an old post, could you please check this :MathRevolution Buneal


Given = (3^x + 1) / 10

we need to find X,

from st1 --- x = 4n+2, n = +ve integer

let n=even, x = Even(Even) + Even = Even
let n=odd, x = Even(Odd) + Even = Even

from this we can conclude x = Even ---> 2,4,6,8........

substitute in given condition :
2 ---> 10/10 = remainder = 0
4 ---> 82/10 = remainder = 2
6 ---> 730/10 = remainder = 0
8 ---> 6563/10 = remainder = 2

from this, the remainder is repeating with a period of 2.

can i say it as sufficient ??


Please shed some light.

Thanks & Kudos in advance

x is a positive integer, is the remainder 0 when \(3^x + 1\) is divided by 10?

Theory: Remainder of a number by 10 is same as the units' digit of the number

(Watch this Video to Learn How to find Remainders of Numbers by 10)

Using Above theory Remainder of \(3^{x} + 1\) by 10 = Unit's digit of \( 3^{x} + 1\)

We know the units' digit of 1, so we need to find the units' digit of 3^x

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

STAT 1: x = 4n + 2, where n is a positive integer.

=> x divided by 4 is same as 4n + 2 divided by 4
=> 4n when divided by 4 will give 0 remainder and 2 when divided by 4 will give 2 remainder
=> Total remainder = 2
=> Units' digit of 3^x will be same as units' digit of 3^2 = 9
=> Units' digit of 3^x + 1 = 9 + 1(=10) = 0
=> Remainder of 3^x by 10 = 1
=> SUFFICIENT

STAT 2: x > 4

=> Now depending of value of x units' digit of 3^x can have different values
=> NOT SUFFICIENT

So, Answer will be A
Hope it helps!

MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem