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value of x

by j_shreyans » Sat Feb 28, 2015 9:57 am
If x is a prime number, what is the value of x?

(1) 2 x + 2 is the cube of a positive integer.

(2) The average of any x consecutive integers is an integer.
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by [email protected] » Sat Feb 28, 2015 11:37 am
Hi j_shreyans,

You might find it easiest to "play around" with this question a bit. Write down a number of possible values for each piece of information and then eliminate possibilities if they DON'T 'match up' with other pieces of information. In this way, the process is similar to TESTing VALUES.

We're told that X is a PRIME number. We're asked for the value of X.

Fact 1: 2X+2 is the CUBE of a positive integer.

Here are the first few positive cubes: 1, 8, 27, 64, 125

Now let's solve for X in each of these scenarios....

2X + 2 = 1
2X = -1 X is negative. Eliminate this option.

2X + 2 = 8
2X = 6
X = 3
3 IS PRIME. X could be 3....

2X + 2 = 27
2X = 25
X = 12.5 This is NOT prime. Eliminate this option.

2X + 2 = 64
2X = 62
X = 31
31 IS PRIME. X could be 31...
Fact 1 is INSUFFICIENT

Fact 2: The average of any X CONSECUTIVE integers is an integer.

This is actually a Number Property rule - it tells us that X MUST be ODD.

Here are some examples.....
2 consecutive integers -- 1, 2 Average = 1.5
3 consecutive integers -- 1, 2, 3 Average = 2
4 consecutive integers -- 1, 2, 3, 4 Average = 2.5
5 consecutive integers -- 1, 2, 3, 4, 5 Average = 3
Etc.

We now know that X is ODD and we know that it's PRIME.
X could be 3, 5, 7, 11,....31
Fact 2 is INSUFFICIENT

Combined (from our prior work) we know that X could be 3 or 31.
Combined, INSUFFICIENT

Final Answer: E

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Rich
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by DavidG@VeritasPrep » Sat Feb 28, 2015 1:23 pm
Rich's explanation is excellent, so I'll just add that the algebra/logic behind the rule underpinning statement 2 is kind of neat.

Say you have 3 consecutive integers. The sum will be x + (x +1) + (x+2) = 3x + 3. The average will be (3x+3)/3 = x + 1 --> integer

Say you have 5 consecutive integers. The sum will be x + (x +1) + (x+2) + (x+3) + (x+4) = 5x + 10. The average will be (5x+10)/2 = x + 2 ---> integer

Now note that you don't even have to do the above calculations if you recall that with evenly spaced sets, the mean =median.

The median for 3 consecutive integers x, (x +1), (x+2) --> (x+1)
The median for 5 consecutive integers x, (x +1), (x+2), (x+3), (x+4) ---> (x+2)

Just another way to help remember the rule: the average of x consecutive integers will always be an integer if x is odd.
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