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DS (Number System)

by rintoo22 » Mon Apr 01, 2013 5:36 am
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.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

I am able to solve the problem, answer E, by plugging in different numbers. But I want a more definitive approach to solve such issues. Can someone please assist ?
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by Anju@Gurome » Mon Apr 01, 2013 6:33 am
rintoo22 wrote: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.
Statement 1: When x is divided by 4, the remainder is (5 - 4) = 1
Hence, when x is divided by 8, the remainder is either 1 or (1 + 4) = 5

Not sufficient

Statement 2: When x is divided by 2, the remainder is 1.
Hence, when x is divided by 8, the remainder is either 1 or (2 + 1) = 3 or (2*2 + 1) = 5 or (2*3 + 1) = 7

Not sufficient

1 & 2 Together: When x is divided by 8, the remainder is either 1 or 5

Not sufficient

The correct answer is E.
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by GMATGuruNY » Mon Apr 01, 2013 7:06 am
A quick lesson on remainders:
When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.

Combined, the statements above imply that when x is divided by 35 -- the LOWEST COMMON MULTIPLE OF 5 AND 7 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.

To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.

Putting it all together:
x = 35c + 18.

Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...

Thus, when x is divided by 33 -- the LCM of 3 and 11 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...
Onto the problem at hand:
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.
Statement 1: x = 12a + 5 = 5, 17, 29, 36...
Statement 2: x = 18b + 11 = 11, 29...

When the statements are combined:
The LCM of 12 and 18 is 36.
The smallest value common to both lists is 29.
Thus:
x = 36c + 29 = 29, 65, 101...

If x=29 is divided by 8, we get:
29/8 = 3 R5.
If x=65 is divided by 8, we get:
65/8 = 8 R1.
Since different remainders are possible, INSUFFICIENT.

The correct answer is E.
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by Brent@GMATPrepNow » Mon Apr 01, 2013 7:14 am
rintoo22 wrote: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.
Another option is to apply a useful rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Okay, now onto the question.

Target question: What is the remainder when x is divided by 8?

Statement 1: When x is divided by 12, the remainder is 5.
Let's list some possible values for x (using the above rule)
Possible values are x = 5, 17, 29, 41 . . . etc
Let's examine 2 possible cases:
Case a: x = 5, in which case the remainder is 5 when x is divided by 8
Case b: x = 17, in which case the remainder is 1 when x is divided by 8
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When x is divided by 18, the remainder is 11.
Let's list some possible values for x (using the above rule)
Possible values are x = 11, 29, 47 . . . etc
Let's examine 2 possible cases:
Case a: x = 11, in which case the remainder is 3 when x is divided by 8
Case b: x = 29, in which case the remainder is 5 when x is divided by 8
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Here we need to find possible values of x that satisfy both statements.
From statement 1, the possible values are x = 5, 17, 29, 41, 53, 65, 77 ...
From statement 2, the possible values are x = 11, 29, 47, 65, 83 ...
One we've already found two possible values of x, let's test them.
Case a: x = 29, in which case the remainder is 5 when x is divided by 8
Case b: x = 65, in which case the remainder is 1 when x is divided by 8
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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