If a, b, and c are consecutive odd positive integers and a

It is worth remembering that there exists only ONE prime number from 90-100 and that is 97. Thus, 91,93 and 95 are not prime.Statement I is definitely not always true.

For a=1,b=3,c=5, Statement II is not true.

As a,b,c are in Arithmetic Progression, the average of the integers\([\frac{(a+b+c)}{3}]\) is equal to the median(b). This is always true.

A.
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Actually this question does not test any obscure knowledge.

The moment you read statement I, you should know that this is not necessarily true. You don't have to test any numbers. This is because you should know a very basic thing about prime numbers - you cannot find a pattern to them i.e. they appear randomly. You cannot say when exactly the next prime number will appear. If it were necessary that out of three consecutive odd numbers, one needs to be prime then there is a pattern to finding them. Hence (I) has to be false.

II. ab>c
This seems would be true often since 5*7 will be greater than 9 etc but what if a is a very small number. So then let's say a = 1.
1*3 < 5
So (II) is not necessarily true.

Then looking at the options III must be true and answer must be (A).

If you want to confirm,
III. a + b + c = 3b
(2X - 1) + (2X + 1) + (2X + 3) = 6X + 3 = 3*(2X + 1)
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I cannot be true in all cases (e.g. 91,93,95 are consecutive odd positive integers but none is a prime).
For II, the case for 1,3,5 does not hold (i.e. for this case, ab<c, whereas ab>c for all other cases)
III must be true since a+b+c= 3b => b = (1/2)*(a+c), which must be true since b, being a consecutive odd integer between a and c, must always be the average of a and c.

A it is.

Do we have to remember such things or is there any way to know that after 90 , there are number not prime.


I wrote till 50 and realized that either of the number was prime and selected "1 " as option.

But 1 is not a prime number. Isn't it?

Yes, 1 is not a prime number. But what this has to do with the question?
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Oh I got it. I used 1, 3, 5 as a test for I and II. That's an error on my part.

The Case 91,93,95 where there are no prime nos. is tested here. as their is only one prime number between 90 to 99 ie 97. And hence option A

115, 117(13*9), 119(17*7) are not prime numbers

Is this a good math?!

Seems like a question testing obscure knowledge that won't be asked for on the GMAT.

OE:

Statement I is generally true for numbers below 100, but once you investigate higher numbers, you find sets of consecutive odd positive integers in which none are prime (for example, 121 (divisible by 11), 123 (divisible by 3), 125 (divisible by 5)). Statement II is true for all sets of consecutive odd positive integers except the lowest, 1, 3 and 5. (1 * 3)<5. Only Statement III is always necessarily true: a is 2 less than b, and c is 2 greater than b. Those differences cancel each other out when you multiply the sum of the set by any positive integer, yielding three times the median. (This property is also related to the rule that, in an evenly spaced set, the median and the mean are the same.)