If m is a positive odd integer between 2 and 30, then m is

No, since 25 (as well as all other possible values of m from statement (1)), is divisible only by one prime number - 5.

If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers?

Since the range is not that big we can write down all possible value of m for each statement.

(1) m is not divisible by 3 --> m can be: 5, 7, 11, 13, 17, 19, 23, 25, and 29. Each has only one distinct prime in its prime factorization. Sufficient.

(2) m is not divisible by 5 --> m can be: 3, 7, 9, 11, 13, 17, 19, 21, STOP. Each but 21 has one prime in its prime factorization, while 21 has two primes: 3 and 7. Not sufficient.

Answer: A.

Hope it's clear.
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The question asks for "... how many different prime numbers?"
If you are thinking that 25 is divisible by 5 and 5, it still counts as one.

(1) m is not divisible by 3
m is not divisible by 2 anyway since we are talking about odd numbers. Next prime number is 5. The next one is 7 but 5*7 = 35 which is greater than 30. So even if we take the two smallest possible primes, no odd number in the given range can have both as factors. So m must have only one prime number as a factor.

(2) m is not divisible by 5
m could be divisible by 3 and 7 (since 3*7 = 21 which lies in our range). So m could be divisible by one prime or by two different primes. So this statement is not sufficient.

Answer (A)
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Sorry Bunuel but I don't quite get it, we are asked it is divisible by how many different positive prime numbers.

So from Statement 1 I understand that it can be divisible by all of them because they are all distinct prime numbers so that will be 9 numbers

But from Statement 2, I don't understand it that well. OK we have 21 that has two primes 3, and 7 but we had already considered them early in the list, so what's the deal with having them again? Since we are asked for different primes then we just ignore these.

Are we being asked to count them all even with repetitions?

Please clarify
Thank you!
Cheers!
J

For (2):

If m=3, then m is divisible by ONE prime factor, 3.
If m=21, then m is divisible by TWO prime factors, 3 and 7.

So, we have two different answers, which means that the statement is not sufficient.

Hope it's clear.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.


If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers?

(1) m is not divisible by 3.
(2) m is not divisible by 5.

In the original condition we have 1 variable and we need 1 equation to match the number of variables and equations. Since there is 1 each in 1) and 2), there is high probability that D is the answer.

In case of 1), m=5(5),25(5),7(7),11(11),13(13),17(17),19(19),29(29) and the different prime factors is 1, therefore the condition is sufficient.
In case of 2), m=3, 15 and since 3 have 1(3), 15 have 2(3,5) prime factors the answer is not unique and therefore not sufficient. Thus the answer is A.

Normally for cases where we need 1 more equation, such as original conditions with 1 variable, or 2 variables and 1 equation, or 3 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore D has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) separately. Here, there is 59 % chance that D is the answer, while A or B has 38% chance. There is 3% chance that C or E is the answer for the case. Since D is most likely to be the answer according to DS definition, we solve the question assuming D would be our answer hence using 1) and 2) separately. Obviously there may be cases where the answer is A, B, C or E.

Hi VeritasKarishma

I am struggling to see what the question is asking

We are being asked per my understanding -- m is divisible by how many different prime numbers ?

Isn't thus the answer to S1 /S2 plausibly -- 1 prime is divisible only / 2 primes are divisible only / 3 primes are divisible only /4 prime numbers are divisible only only .......n prime numbers are divisible only

If i solve for S1 -- i see 9 different prime numbers possibly [5 / 7 / 11 / 13 / 17 / 19 / 23 / 25 (5x5 is still one prime number only, i.e. 5) / 29 ]

I marked S1 as sufficient because i got 9 different prime numbers

Is my answer to S1 (=9) incorrect ?

--------------------------------
On the other hand my answer to S2 is 9 as well

S2 set : 3 / 7 / 9 / 11 / 13 / 17 / 19 / 23 / 25(5*5) / 29

prime numbers = 3 / 7 / 11 / 13 / 17 / 19 / 23 / 5 / 29

Please let me know, where is the break in my logic

I don't think you have understood the question. Let me break it down:

Given: "m is a positive odd integer between 2 and 30."
Conclusion: m can be 3/5/7/9/11/13... 29

Question: m is divisible by how many different positive prime numbers?
Answer; Depends on the value of m.
If m is 3, answer is that m is divisible by 1 prime number.
If m is 5, answer is that m is divisible by 1 prime number.
...
If m is 15, answer is that m is divisible by 2 prime numbers.
...

Let's look at the stmnts that will give us more info about m.

(1) m is not divisible by 3.
- m could be 5 in which case it has 1 prime factor.
- m could have 2 prime factors the smallest acceptable ones being 5 and 7. But that would mean m = 5*7 = 35 but m must be less than 30. So m cannot have 2 prime factors and then obviously, not more than 2 either.
Hence m must have only 1 different positive prime factor.
Sufficient

(2) m is not divisible by 5.
- m could be 3 in which case it has 1 prime factor.
- m could be 21 in which it has 2 prime factors (3 and 7)
Not sufficient.

Answer (A)
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