What is the remainder when the positive integer x is divided

Question

What is the remainder when the positive integer x is divided by 8?

(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

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When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers :oops:

)

(I picked numbers :oops:

)

kevincan · Answer

Picking numbers is a great way to go!

Clearly, neither is sufficient

We know that x=12k+5 so that x â‚¬ {5,17,29,42,53,65...}

Also x=18m+11 so that xâ‚¬ {11,29,47,65...}

29 and 65, both possible values of x, yield different remainders

IanStewart · Answer

In many remainders questions, it's enough just to find a couple of numbers that 'work' with the given information, and if you simply list the first few numbers that satisfy each statement, it's easy to judge if the statements together are sufficient:

1) x = 5, 17, 29, 41, 53, 65, 77, 89 ...
2) x = 11, 29, 47, 65, 83, 101 ...

Since 29 and 65 give different remainders when you divide by 8, the answer is E.

More abstractly, when combining two statements like the above, the pattern will be based on the LCM of the two divisors. Here, we can consider dividing x by 36 = LCM(12, 18).

Notice that, if Statement 1 is true, the remainder will be 5, 17 or 29 when x is divided by 36.
If Statement 2 is true, the remainder will be 11 or 29 when x is divided by 36.
If both Statements are true, the remainder therefore must be 29 when x is divided by 36.

That's still not sufficient, as above; x could be 29, or x could be 65.

KarishmaB · Answer

Let's look at the question:

What is the remainder when the positive integer x is divided by 8?
This means: What is leftover when you make groups of 8?

Statement 1: When x is divided by 12, the remainder is 5.

When you make groups of 12, 5 balls are leftover. When you make groups of 8 instead, each of the groups of 12 balls leaves 4 balls. If no. of groups of 12 is even, you can combine 2 groups of 4 balls each to make more groups of 8. In that case, 5 balls will be still leftover. So a remainder of 5 is possible.

If no. of groups of 12 is odd, 4 balls will be leftover from one group of 12 and 5 balls will be still leftover. So a total of 9 balls will be leftover. We can make another group of 8 out of these 9 balls and 1 ball will be leftover. So a remainder of 1 is also possible.

Since remainder can be 5 or 1, this statement alone is not sufficient.

Statement 2: When x is divided by 18, the remainder is 11.

When you make groups of 18, 11 balls are leftover. When you make groups of 8 instead, each of the groups of 18 balls makes 2 groups of 8 balls each and leaves 2 balls. 
Now there are 4 possibilities:
1. We are left with 2 balls + the original 11 remaining balls = 13 balls
When you make another group of 8 from 13, remainder will be 5
2. We are left with 2+2 balls + the original 11 remaining balls = 15 balls
When you make another group of 8 from 15, remainder will be 7
3. We are left with 2+2+2 balls + the original 11 remaining balls = 17 balls
When you make 2 groups of 8 from 17, remainder will be 1
4. We are left with no groups of 2 balls since they all make a complete group of 8. Only the original 11 balls are remaining. When you make a group of 8 from 11, remainder will be 3.

Since remainder can be 5, 7, 1 or 3, this statement alone is not sufficient.

Using both statements, remainder can be either 5 or 1 so they both together are not sufficient.
Answer (E)

abicool456 · Answer

Where are the answer options?

Bunuel · Answer

Hi, and welcome to GMAT Club.

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post  ALL YOU NEED FOR QUANT .

Hope this helps.

Icecream87 · Answer

Is my way of doing this correct?:

(1)  12q + 5 = x  insuf

(2)  18b + 11 = x  insuf

Together:  
 12q + 5 = 18b +11

12q - 18b = 6

6(2q - 3b) = 6

This leads us to  2q - 3b = 1

I replace q by b in equation 1:

12(\frac{1 + 3b}{2}) +5 = x

18b + 11 = x

Statement 1 becomes equal to statement 2 : Hence insufficient.

Icecream87 · Answer

I did it this way:

(1)  12q + 5 = x  insuf

(2)  18b + 11 = x  insuf

Together:  
 12q + 5 = 18b +11

12q - 18b = 6

6(2q - 3b) = 6

This leads us to  2q - 3b = 1

I replace q by b in equation 1:

12(\frac{1 + 3b}{2}) +5 = x

18b + 11 = x

Statement 1 becomes equal to statement 2 : Hence insufficient.

abhimahna · Answer

Option 1 : Possible Numbers could be 5,17,29,41,53,65,...

when divided by 8, they will give remainders are 5,1,5,1,...

Hence, not sufficient.

Option 2 : Possible Numbers could be 11,29,47,65,...

when divided by 8, they will give remainders are 3,5,7,5,...

Hence, not sufficient.

Combining both the statements, Possible no. could be 29 and 65.

When divided by 8,They will give remainders as 5,1.

Hence, after combining also, it is insufficient. Correct Answer : E.

Bambi2021 · Answer

Great approach by  IanStewart  here, to realize that the LCM will be the leap between possible values of x.

I just thought that:

x = 12p + 5 = 18k + 11
12p = 18k + 6
If p=2, then k=1 and x = 29.
If p=5, then k=3 and x = 65.

But I had to match every value of p with k. Realizing that the values will be LCM (in this case 36) apart, solving it gets much faster.

lucasd14 · Answer

Hi  KarishmaB , by this approach, how did you conclude that 5 and 1 would be the possible remainders when checking both eq. together? Thanks.

KarishmaB · Answer

lucasdachequi

Statement 1 tells us that remainder can be 5 or 1.

Statement 2 tells us that remainder can be 5, 7, 1 or 3.

If I use both statements, i.e. both must hold, still I see that values 5 and 1 are common to them. I can ignore 7 and 3 now since statement 1 does not satisfy them but 5 and 1 are common to both statements. So even after using both statements, still 2 values are possible fro remainder. But what we need is one unique value. 
Hence, answer is (E)

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What is the remainder when the positive integer x is divided

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