The concept covered in this question is tested quite often in the real GMAT exam. 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.

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 a leap year or the meaning of the word counterclockwise), you must indicate whether -

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

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 5: **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.

INR

**Q1. 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.

**Q2. 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 given in the statements is not sufficient to answer the question.

**Q3. What information do we have about x from the question stem?**

x is a positive integer.

**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. 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.

**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.

We are **not able to get a unique remainder** using statement 1.

Hence, statement 1 is not sufficient.

__Eliminate answer options A and D__.

**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. Hence, 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 k varies.

**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.

We are **not able to get a unique remainder** using statement 2.

Hence, statement 2 is not sufficient.

__Eliminate answer option B__.

**Statements**: When x is divided by 7, the remainder is 5 & 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.

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.

**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.

We are **not able to get a unique remainder** despite combining the two statements, the data provided is NOT sufficient.

Hence, statements together are not sufficient.

__Eliminate answer option C__.

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