Remainders - Number Properties : Data Sufficiency

The concept tested in this question is tested quite often in the GMAT. Determining the remainder when a number is divided by a divisor - data provided will be the remainders when the same number is divided by two other divisors. One could solve these questions by framing equations or substituting values. Bear in mind, you should substitute values to prove insufficiency. You may not be able to conclusively prove sufficiency by susbtituting numbers.

Directions

• DS
• This 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.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question

What is the remainder when the positive integer x is divided by 6?
Statement 1: When x is divided by 7, the remainder is 5.
Statement 2: When x is divided by 9, the remainder is 3.

Choice E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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Detailed Solution

1. What kind of an answer will the question fetch?

The question asks us to find the remainder when x is divided by 6.
The data provided in the statements will be considered sufficient if we get a unique value for the remainder.

2. When is the data not sufficient?

If after using the information given in the statements, we are not able to determine a unique remainder when x is divided by 6, the data statements is not sufficient to answer the question.

3. What information do we have about x from the question stem?

x is a positive integer.

Now let us evaluate each statement independently.

Statement 1: When x is divided by 7, the remainder is 5.

Approach 1

x can therefore, be expressed as 7k + 5

If k = 0, x = 5. Hence, the remainder when x is divided by 6 will be 5.
If k = 1, x = 12. The remainder when x is divided by 6 will be 0.

The remainder varies as the value of k varies.

Because we are not able to get a unique remainder using statement 1, statement 1 is NOT Sufficient.

Approach 2

List numbers (about 10 to 12) that satisfy the condition given in statement 1 and check whether you get a unique remainder

x could be 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89 .....

The remainders that we get correspondingly are 5, 0, 1 and so on

As seen with Approach 1, we are not getting a unique remainder. Hence, statement 1 is NOT sufficient.

Therefore, we can eliminate choices A and D. We can narrow down the choices to this DS question to B, C, or E.

Statement 2: When x is divided by 9, the remainder is 3.

Approach 1

x can be expressed as 9p + 3

If p = 0, x = 3. The remainder when x is divided by 6 will be 3.
If p = 1, x = 12. The remainder when x is divided by 6 will be 0.

The remainder varies as the value of p varies.

Because we are not able to get a unique remainder using statement 2, statement 2 is also NOT Sufficient.

Approach 2

List numbers (about 10 to 12) that satisfy the condition given in statement 2 and check whether you get a unique remainder

x could be 3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 .....

The remainders that we get correspondingly are 3, 0, 3 and so on

As seen with Approach 1, we are not getting a unique remainder. Hence, statement 2 is NOT sufficient.

Therefore, we can eliminate choice B as well. We are down to C or E.

Combining the two statements

When x is divided by 7, the remainder is 5 and when x is divided by 9 the remainder is 3.

Approach 1

Equating information from both the statements, we can conclude that 7k + 5 = 9p + 3.

Or 7k = 9p - 2.

i.e., 9p - 2 is a multiple of 7. When p = 1, 9p - 2 = 7. So, one instance where the conditions are satisfied is when k = 1 and p = 1.
x will be 12 and the remainder when x is divided by 6 is 0.

If we can find another value that satisfies the condition that 7k = 9p - 2 and we get a different remainder, the data is NOT sufficient. If we get the same remainder for all values that satisfy this condition, then the data is sufficient.

When p = 2, 9p - 2 is not divisible by 7. Proceeding by incrementing values for p, when p = 8, 9p - 2 = 70, which is divisible by 7.

When p = 8, x = 75.

The remainder when 75 is divided by 6 is 3

The remainder when x was 12 was 0. The remainder when x is 75 is 3

Despite combining the two statements, we are still not getting a unique remainder when x is divided by 6, the data provided NOT Sufficient.

Approach 2

The values of x that satisfy statement 1 are 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89 ...

The values of x that satisfy statement 2 are 3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 .....

Values of x that are present in both sets are 12, and 75 from the set listed above.

The remainders when 12 and 75 are divided by 6 are 0 and 3 respectively.

Because we do not have a unique remainder despite combining the two statements, the data provided is NOT sufficient.