i think you misunderstood it.

what it is saying you is
The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.

meaning integer like 10 where a = 1 and b=0; 11 where a =1, b=1 ; 12 where a = 1 and b =2
(note that it reads 2 digit number 'ab' and is not a * b)

so basically all the numbers that satisfy condition 1 will be 12, 15, 18, 21, 24, 27, 30, 33, 36, 39....and so on
notice that are multiple of 3.

now a+b for all these numbers = 3, 6, 9, 3, 6, 9, 3, 6 follow the patter of 3, 6 , 9 and are multiple of 3.

1) is also the check for 'division by 3'.

I.e. a number is a multiple of 3 if its digits added together are divisible by 3. Therefore, if ab is divisible by 3, then a+b must be divisible by 3.

Tough and Tricky questions: Multiples.

If a and b are both single-digit positive integers, is a + b a multiple of 3?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.
(2) a – 2b is a multiple of 3.
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D.

1) ab is a multiple of 3.
By basic rule of division, a number is div by 3 if the sum of its digits is div by 3.
Reason: 10a+b mod 3 = 0
=> (a+b) mod 3 = 0 [since 10 mod 3 = 1]
so sufficient.

2) a-2b = 3k
a = 3k+2b
=> a+b = 3k+2b+b = 3(k+b) which is div by 3
so sufficient.
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a = 3k + 2b;

Add b to both sides: a + b = (3k + 2b) + b;

a + b = 3k + 3b;

a + b = 3(k + 3).
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Before solving the question, you need some knowledge. “The remainder of a certain positive integer divided by 3 and 9 is same as the sum of every digit place of n divided by 3 and 9.
There are 2 variables in the original condition. In order to match the number of variables to the number of equations, 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:
Condition 1) – a multiple of ab=3 is a multiple of a+b=3. Hence, the answer is yes and the condition is sufficient.
Condition 2) – If a+b is a multiple of a-2b=3, then, it becomes a multiple of a-2b+3b=3 plus 3b and a multiple of a+b=3 plus 3b=3(multiple+b)=3. Hence, the answer is yes and the condition is sufficient. Thus, the correct answer is D.
This is 50-51 level question. Please remember the common mistake type 4(B). If the answer is too easily A or B, please consider D.

- For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

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