If a and b are integers, what is the value of a + b ?

I would go with E.

Since non of the options tell if a=b

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There are 2 variables in the original condition (a and b). In order to match the number of equations and the variables, we need 2 equations. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer. Using the condition 1) and the condition 2) at the same time, C is the correct answer. However, this is an integer question, one of the key questions. Hence, we can apply the Common Mistake Type 4(A). Using the condition 2) alone, we get a=495 and b=496. The answer is unique and the condition is sufficient. The correct answer is B. Hence, if C and B are both the answers, B is the correct answer.

Statement 1 tells us 493 < a < b < 498
As per this statement a can be 494,495,496 whereas b can be 495,496,497.Following the condition a<b, we get different values a+b depending upon which values we choose so insufficient
Statement 2 tells us 494 < a < b < 497
As per this statement a can be only 495 and b can be only 496. so we get a definite value for a+b. So sufficient
Correct Answer - B