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.
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.
x is a positive integer.
Now let us evaluate each statement independently.
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.
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.
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.
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.
When x is divided by 7, the remainder is 5 and when x is divided by 9 the remainder is 3.
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.
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.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Choice E is the correct answer.