Question: A certain fruit stand sold a total of 76 oranges to 19 customers. How many of them bought only one orange?

1. None of the customer bought more than 4 oranges
2. The difference between the number of oranges bought by any two customers is even

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution and Explanation:

Approach Solution (1):

S1: None of the customers bought more than 4 oranges. This basically means that all customers bought exactly 4 oranges (76/19 = 4), because if even one customer bought less than 4, the sum would be less than 76. Hence, no one bought only one orange.
Sufficient

S2: The difference between the numbers of oranges bought by any two customers is even. In order for the difference between any numbers of oranges bought to be even, either all customers must have bought an odd number of oranges or all customers must have bought an even number of oranges. But the first case is not possible: the sum of 19 odd numbers is odd and not even like 76.
Hence again no one bought only one (i.e. odd number) orange.
Sufficient

Correct Option: D

Approach Solution (2):

S1: None bought < 4 – clearly sufficient since then everyone bought 4 oranges
S2: Everyone bought something so there is no customer who has 0 orange

Moving ahead, if 18 customers each bought 1 and 19th customer bought 76 – 18 = 58 now if we take difference of any 2 numbers from it such as 1 – 1 = 0 or 58 – 1 Not valid
Giving 2 oranges to each of the customers, hence 18 customers have 2 and 18th one has 76 – 36 = 40 oranges, now difference of any number in the set is even. It means none of them had 1 orange
Sufficient

Correct Option: D

Approach Solution (3):

Statement 1: None of the customers bought more than 4 oranges.
The key thing to notice here is that 76 = 19 x 4.
So if 4 is the maximum anyone bought, then the only way that the total would be 76 is by all of them buying 4. If any bought fewer than 4, the total would be less than 76.
So none bought only 1.
Sufficient.

Statement 2: The difference between the number of oranges bought by any two customers is even.
To get the answer to this one you could start off by wrapping you mind around what is going on by using what we figured out from Statement 1, that they all bought 4. If they all bought 4, then the difference between the numbers of oranges bought by any two is 0, an even number.
So one possibility is that they all bought 4, and none bought 1.

Could some have bought 1?
If everyone were to have bought an odd number of oranges, the difference between any two numbers bought would be even. Odd - Odd = Even.
So at first it seems that the numbers bought could be all even or all odd.
However, there are 19 people. 19 is odd, and the sum of an odd number of odd numbers is always odd. So there is no way that the sum of an odd number of odd numbers could be 76, an even number.

So everyone must have bought an even number of oranges, meaning nobody bought 1 orange.
Sufficient.

Correct Option: D

“A certain fruit stand sold a total of 76 oranges to 19 customers. How many of them bought only one orange?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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