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.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer thequestion asked;
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer thequestion asked;
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked,but NEITHER statement ALONE is sufficient;
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 are needed.
Correct answer is option 'E'. Can you explain this answer?

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.

a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer thequestion asked;
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer thequestion asked;
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked,but NEITHER statement ALONE is sufficient;
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 are needed.

Correct answer is option 'E'. Can you explain this answer?

Answer

Step 1: Decode the Question Stem and Get Clarity
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.

Step 2: Evaluate Statement 1 ALONE
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.

Step 3: Evaluate Statement 2 ALONE
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.

Step 4: Evaluate Statements TOGETHER
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.

Choice E is the correct answer.

Answer

Statement 1: When x is divided by 7, the remainder is 5.
Statement 2: When x is divided by 9, the remainder is 3.

To find the remainder when x is divided by 6, we need to determine the possible values of x that satisfy both statements.

Analysis of Statement 1:
If x leaves a remainder of 5 when divided by 7, the possible values of x are: 5, 12, 19, 26, 33, ...

Analysis of Statement 2:
If x leaves a remainder of 3 when divided by 9, the possible values of x are: 3, 12, 21, 30, 39, ...

Combining the Statements:
To satisfy both statements, x must be a value that appears in both lists. The only value that satisfies both lists is 12. Therefore, x = 12.

Conclusion:
We have a unique value for x that satisfies both statements, so we can determine the remainder when x is divided by 6. When x = 12, the remainder is 0 when divided by 6.

Answer:
Both statements together are sufficient to answer the question asked, and the answer is not E.

Answer