If x is a positive integer, is the remainder 0 when 3^x + 1 is divided

Question

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

(1) x = 4n + 2, where n is a positive integer.
(2) x > 4

Bunuel · Accepted Answer

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

(1) x = 4n + 2, where n is a positive integer.

Last digit of  3^x  repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4. Now,  3^{4n+2}  will have the same last digit as  3^2  (remainder upon division 4n+2 upon cyclicity 4 is 2, which means that 3^{4n+2} will have the same last digit as 3^2). Last digit of  3^2  is  9 . So  3^{4n+2}+1  will have the last digit  9+1=0 . Number ending with 0 is divisible by 10 (remainder 0). Sufficient.

(2) x > 4. Clearly insufficient.

Answer: A.

adiagr · Answer

This question checks divisibility of 3. 
 
 Concentrate only on last digit of product and not the entire value.

Now  3^4 = 81, i.e. last digit is  1 .  3^5  has last digit  3 .

So  3^{4x}  will have  "1"  as its last digit.  3^{4x}  can be written as ( 3^4 )^x

Statement 1 
 3^{4n+2}  is nothing but ( 3^{4n} )* ( 3^2 ). The last digit of this product will be  9 .
 
Adding 1 to this sum will yield a zero  and so the entire product  will be  divisible by 10. 
 
(1) is sufficient  
 
 Statement 2

x>4. 
  x=5  --> This will have last digit of 3's power as 3  3^4 )*(3) => Last digit will be 3]. 
Add 3+1 = 4.  Not  divisible by 10. 
 
  x=6  --> This will have last digit of 3's power as 9  3^4 )*(9) => Last digit will be 9]. 
Add 9+1 = 10.  Will be   divisible by 10.

So Statement (B) is  not sufficient .

Answer is  A

EgmatQuantExpert · Answer

Though this is quite an easy question, the accuracy statistics suggest that roughly 1 out of every 4 students who attempted this question got it wrong. This may have happened because the students first straight-away started from the given statements and then, got confused in processing the question statement and the Statement 1 (and 2) together. Here's an alternate solution that eliminates the chances of such confusion! Let's first analyze the question statement alone: The given expression is 3^{x} + 1 The units digit of 3 can be 3 (for powers of the form 4m+1), 9 (for powers of the form 4m+2), 7 (for powers of the form 4m+3) or 1(for powers of the form 4m) Out of these 4 possible unit digits, the sum 3^{x} + 1 will be divisible by 10 only when the units digit is 9. So, the question is actually asking us to find if the power of 3, that is x, is of the form 4m + 2 or not. Now that we've simplified the question, analyzing the 2 statements is going to be a cakewalk! :-D

Please note how the analysis in my solution is quite similar to the analysis in the solutions posted above. The point of difference comes in when I did that analysis. I did it before I went to Statements 1 and 2. The benefit of doing this analysis with the question statement itself is that I now have a very clear idea of what I need to look for, in order to answer the question. And because of this clear idea, the chances of my getting confused by irrelevant information in Statements 1 and 2 are also greatly reduced. Hope this helped!

Japinder

golfwhiskey · Answer

this has to be pre-taught knowledge right? who th can just sit there and figure out in a minute that exponents of 3 cycles through just 4 ending numbers and in repeating order, which one of them is the ending you need. so a formula of multiples of 4 and taking the second number in this 4 number sequence gets you the exponent you need
???

Sa800 · Answer

this def has to be known before. how can you just randomly come up with the cyclicity rule of the power of 3? Does GMAT like these sorts of cyclicity questions?

Also another question...why are we solely looking at the units digits?

Bunuel · Answer

When you divide a positive integer by 10, the remainder is always the unit's digit of that integer. For instance:

1 2  when divided by 10 gives the remainder of  2 ; 
3 0  when divided by 10 gives the remainder of  0 ; 
234 4  when divided by 10 gives the remainder of  4 ; 
...

Check other  Units digits, exponents, remainders problems directory  in our  Special Questions Directory .

Hope it helps.

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=h1HTEhmUb1w

ManifestDreamMBA · Answer

For  3^x + 1  to have remainder 0, when divided by 10, it must end in 0
3^1 = 3+1 = 4
3^2 = 9+1 = 1 0 
3^3 = 27+1 = 2 8 
3^4 = 81+1 = 8 2

Only power of 3^2 will result in a number ending with 0

If x = 4n+2, then it must end in 0

Statement 1
It is aligned with what we are looking for
Sufficient

Statement 2
x>4
It could be 5,6,7,8 leading to  3^x + 1  ending in 4,0,8,2 respectively
Insufficient

Answer A

If x is a positive integer, is the remainder 0 when 3^x + 1 is divided

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