What is the remainder when the positive integer x is divided by 3 ?

Question

(1) When x is divided by 6, the remainder is 2.
(2) When x is divided by 15, the remainder is 2.

Bunuel · Accepted Answer

What is the remainder when positive integer x is divided by 3?

(1) When x is divided by 6, the remainder is 2 --> \(x=6q+2=3(2q)+2\) --> the remainder upon division x by 3 is 2. Sufficient.

(2) When x is divided by 15, the remainder is 2 --> \(x=15p+2=3(5p)+2\) --> the remainder upon division x by 3 is 2. Sufficient.

Answer: D.

For more on remainders check here: remainders-144665.html

Hope it helps.

EMPOWERgmatRichC · Answer

Hi All,

These types of 'remainder' questions are almost always easily solved by TESTing VALUES.

We're told that X is a positive integer. We're asked for the remainder when X is divided by 3.

Fact 1: When X is divided by 6, the remainder is 2

I'm going to list out the first few integers that fit this description:

X = 2, 8, 14, 20, 26, 32, etc....

The pattern here is that each number is "6 more" than the one before it. Now, let's see what happens when we use these values in the question:

IF...
X = 2
2/3 = 0 remainder 2

X = 8
8/3 = 2 remainder 2

X = 14
14/3 = 4 remainder 2

X = 20
20/3 = 6 remainder 2
Etc.

The pattern here is clear (and you could probably name the next few "results" without doing any calculations at all). The remainder is ALWAYS 2.
Fact 1 is SUFFCIENT

Fact 2: When X is divided by 15, the remainder is 2

Here are the first few terms that fit this Fact:

X = 2, 17, 32, 47, etc.

IF....
X = 2
2/3 = 0 remainder 2

X = 17
17/3 = 5 remainder 2

X = 32
32/3 = 10 remainder 2

X = 47
47/3 = 15 remainder 2
Etc.

Just as in Fact 1, we have a clear pattern here. The answer is ALWAYS 2.
Fact 2 is SUFFICIENT

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

mau5 · Answer

From F.S 1, we have that x = 6k+2, where k is a non-negative integer constant. Required to find x = 3p+r, where p= again a non-negative integer constant. We can see that for some value, 6k = 3*(2k)  = 3p. Thus, the remainder when divided by 3 will also be 2. Sufficient.

Similarly, from F.S 2 , we have that x = 15t+2. Just as above, for some integer, 15t =  3*(5t)  = 3p. Thus, the remainder is 2.Sufficient.

D.

manimgoindown · Answer

Thanks even though it's my off day I'm going to run through that thread. Thanks for sorting the problems by difficulty. Your work and its contribution has been immense to my preparation Bunuel

Bunuel · Answer

What is the remainder when positive integer x is divided by 3?

(1) When x is divided by 6 remainder is 2 --> x = 6q + 2 = {multiple of 3} + 2, thus the remainder when x is divided by 3 is 2. Sufficient.

(2) When x is divided by 15, remainder is 2 --> x = 15p + 2 = {multiple of 3} + 2, thus the remainder when x is divided by 3 is 2. Sufficient.

Answer: D.

Hope it's clear.

sauberheine1 · Answer

I am a little bit confused even though its just a simple problem.

My remainder formula is : x/y = x/y + r/y where R is the remainder ...

any one able to help me out ? Sorry !

Bunuel · Answer

Check here: remainders-144665.html If x and y are positive integers , there exist unique integers q and r , called the quotient and remainder , respectively, such that y =divisor*quotient+remainder= xq + r and 0\leq{r}

adkikani · Answer

EMPOWERgmatRichC   niks18   chetan2u   amanvermagmat   gmatbusters   KarishmaB

Do you have any strategy for picking numbers? I tried two random numbers when I got this Q in my mock and was lucky to get through.

niks18 · Answer

Hi adkikani the algebraic approach has already been explained by Bunuel here https://gmatclub.com/forum/what-is-the- ... l#p1192221

KarishmaB · Answer

Number picking is not a good idea here. If you do want to use it, you need to find the repeating pattern. Here, every number you take that satisfies either stmnt, you will get the same answer. But say, stmnt 1 was (1) When x is divided by 5, the remainder is 2. So in case of 7, when you divide by 3, you will get 1. In case of 12, when you divide by 3, you will get 0. In case of 17, when you divide by 3, you will get 2. In case of 22, when you divide by 3, you will get 1. In case of 27, when you divide by 3, you will get 0. so you see a pattern and all your cases are covered. Much better would be to look at this from a holistic angle. Check here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... unraveled/

visheshsahni · Answer

Hello  KarishmaB , as a general rule or with practise, I don't even need to find a pattern, I would try to save as much time as possible. If the question would have been a little different suppose, I wanted remained when same number is divided by 11.

I can see when divided by 6 leaves a remainder of 2, when divided by 15 same remainder. i would just take LCM of both and add the remainder which will give me value of N.

LCM is 30 and N is 30+2. This came to me with practise. So coming back to divisibility by 3. Rem is 2 , by 7 it would be 4.

Pattern approach is important to come to understanding my approach. Kudos for explaining basics.

Kinshook · Answer

Asked: What is the remainder when the positive integer x is divided by 3 ?

(1) When x is divided by 6, the remainder is 2.
x= 3*2k+ 2 = 2 mod 3
SUFFICIENT

(2) When x is divided by 15, the remainder is 2.
x= 3*5k+2 = 2 mod 3
SUFFICIENT

IMO D

Posted from my mobile device

TheNightKing · Answer

Try calculating remainder using numbers of the given form in each statement:
Statement 1: 
X = 2
2/3 => R= 2
X = 8
8/3 => R= 2
X = 14
14/3 => R= 2

Sufficient.

Statement 2:
X = 2
2/3 => R= 2
X = 17
17/3 => R= 2

Sufficient.

Hence D

myhigheredmba · Answer

In response to method explained by  Bunuel  ( Any expert can answer! ) - True or false - If statement two had said:

(2) When x is divided by 14, the remainder is 2.

^ This would be insufficient to answer the question: What is the remainder when the positive integer x is divided by 3,  correct??

TheNightKing · Answer

Question: Remainder when x divided by 3?

Statement : (2) When x is divided by 14, the remainder is 2.

Firstly you are picking a composite number so it becomes tricky. I will ignore it for a moment.

2/14 works. 2/3 -> R=2

16/14 works. 16/15 -> R=1

So Not Sufficient.

myhigheredmba  I don't see any reason why you should do this.?

chloe2m · Answer

Hi  Bunuel , 
I understand the reasoning, but I keep having the same reflex where I would write : 
As we know that x = 6q + 2 then x / 3 = (6q + 2)/3 = 2q + 2/3 
And I would assume that remainder is  2/3  where in fact it is  2 . 
Could you please advise me with what is wrong with that logic ?

Thank you for your help

Bunuel · Answer

Positive integer  a  divided by positive integer  d  yields a reminder of  r  can always be expressed as  a=qd+r , where  q  is called a quotient and  r  is called a remainder, note here that  0\leq{r}<d  (remainder is non-negative  integer  and always less than divisor).

Thus, a remainder must be a non-negative  integer ; it cannot be a fraction.

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What is the remainder when the positive integer x is divided by 3 ?

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