What is the remainder when the positive integer X is divided by 12?

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For remainder questions, you solve such questions by: direct substitution  find the first number that two numbers have in common as multiples  add the least common multiple of the numbers you divide by.

There is 1 variable (x) in the original condition. In order to match the number of variables to the number of equations, we need 1 equation. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that D is the correct answer.

In the case of the condition 1), if we directly substitute, we get x=3p+2=2,5,8,11… Then, 2=12*0+2-->2, 5=12*0+5-->5. The answer is not unique and the condition is not sufficient.
In the case of the condition 2), if we directly substitute, we get x=4q+3=3,7,11..... Then, =12*0=3-->3, 7=12*0+7-->7. The answer is not unique and the condition is not sufficient.

Using the condition 1) and the condition 2), we find the x=11 (the first the number that two have in common as multiple). Then, we add the least common multiple of 3 and 4, which is 12. Then we get x=11, 23, 35,..... Since =12*0+11, 23=12*1+11, 35=12*2+11, we always get 11. The answer is unique and the condition is sufficient. Therefore, the correct answer is C.

Here is a post discussing how to solve such questions:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/05 ... s-part-ii/

It is obvious that neither stmnt alone can be sufficient since we are considering divisibility by 12. We need to find the divisibility by both 3 and 4.

Stmnt 1: X = 3a + 2
Stmnt 2: X = 4b + 3

Try b = 0, 1 , 2... When b = 2, X = 11 which is also of the form 3a + 2.
So X can take the smallest value of 11.
Next values of X will be 12n + 11 where n is a positive integer. In all these cases, the remainder will be 11.

Answer (C)

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