Nice explanation, vineeshp!
One thing to add here - I'd consider this to be in the number properties / divisibility family, and for a lot of these problems there's a 2-part way to solve them and learn about them.
1) Find a pattern in the numbers that satisfy the condition, and see if that pattern is enough to solve. That's what vineeshp did here by identifying numbers that work for each statement and recognizing that they all provide a remainder of 3 when divided by 6.
2) Determine why that pattern holds by analyzing the numbers and how they react. Step 1 is often all you need, particularly on a timed test, but Step 2 is a great way to really understand the GMAT while you're studying. Let's break these down:
Statement 1: A number that has a remainder of 1 when divided by 2 is an ODD number. And the remainder of 0 when divided by 3 means that it's divisible by 3. So we're looking for an odd multiple of 3 here. Only even multiples of 3 are divisible by 6; every odd multiple of 3 is 3 places away from a multiple of 6, so the remainder will always be 3. That's why we know for certain that this is sufficient.
Statement 2: Here, 12 is a multiple of 6, so if we're always 3 places away from being divisible by 12, we're also always 3 places away from being divisible by 6. Again, this means that the remainder will always be 3.
Overall, my point is that, in practice at least, if you look for the underlying patterns behind the sets of numbers that a statement or sequence provides, you can start to get pretty quick at identifying patterns and proving answers. And since the GMAT loves to test patterns, number properties, divisibility, etc., training yourself to think that way will provide a huge advantage for you.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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