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

[hazelnut01]

If a, b, and c are consecutive odd positive integers and a<b<c, which of the following must be true?

I. at least one of the three numbers is prime
II. ab>c
III. a + b + c = 3b

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Official answer = A
Source = Veritas Prep

[Jay@ManhattanReview]

If a, b, and c are consecutive odd positive integers and a<b<c, which of the following must be true?

I. at least one of the three numbers is prime
II. ab>c
III. a + b + c = 3b

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Official answer = A
Source = Veritas Prep

Hi ziyuenlau,

Let us test each statement one by one,.

S1: At least one of the three numbers is prime

One must know prime numbers till 100. Between 90 to 100, there are only one prime number '97.' So, there can be three consecutive odd positive integers such that none is prime: 91, 93, and 95. This statement must not be true.

S2: ab > c

Say, the three consecutive odd positive integers are 1, 3, and 5. Thus, 1*3 = 3 < c = 5. This statement must not be true.


S3: a + b + c = 3b

This statement must be true.

For any three consecutive positive integers, the middle-most integer, here b, is the mean of the three consecutive positive integers.

Thus, (a + b + c)/3 = b

=> a + b + c = 3b

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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Manhattan Review GMAT Prep

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[GMATGuruNY]

If a, b, and c are consecutive odd positive integers and a<b<c, which of the following must be true?

I. at least one of the three numbers is prime
II. ab>c
III. a + b + c = 3b

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

While it is possible under timed condtions to prove that a huge integer is NOT prime -- if the integer is even, if the integer is a multiple of 3, etc. -- it is NOT possible under timed conditions to prove that a huge integer IS prime.
Thus, if a GMAT problem asks whether a huge integer is prime, the answer must be NO.

I: at least one of the three numbers is prime
Here, it is possible that a, b and c are HUGE.
Since the problem asks whether these potentially huge numbers are all prime, the answer must be NO.
Since Statement I does not have to be true, eliminate any answer choice that includes Statement I.
Eliminate B, C and E.

II: ab > c
If a=1, b=3, and c=5, then ab < c.
Since Statement II does not have to be true, eliminate any remaining answer choice that includes Statement II.
Eliminate D.

The correct answer is A.