If x is a positive integer, is x

Question

If x is a positive integer, is x<16?

(1) x is less than the average (arithmetic mean) of the first ten positive integers
(2) x is the square of an integer.

Kurtosis · Accepted Answer

St1: Average of first 10 positive integers = Sum/10 = ((10/2)(1 + 10))/10 = 55/10 = 5.5
x < 5.5 --> Sufficient

St2: x = Integer^2 = 1, 4, 9, 16, 25 
x can be less than 16 or greater than 16 --> not sufficient

Answer: A

BrentGMATPrepNow · Answer

Target question :    Is x < 16?

Statement 1 : x is less than the average (arithmetic mean) of the first ten positive integers  
Since the first ten positive integers are EQUALLY SPACED, the average = (smallest number + biggest number)/2 = (1 + 10)/2 = 5.5
So, x < 5.5
Since 5.5 < 16, we can also write: x < 5.5 < 16
From here, we can see that the answer to the target question is  YES, x IS less than 16 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : x is the square of an integer. 
There are infinitely many values of x that satisfy statement 2. Here are two: 
 Case a : x = 4 (4 is a square of an integer, since 4 = 2²). In this case, the answer to the target question is  YES, x IS less than 16 
 Case b : x = 25 (25 is a square of an integer, since 25 = 5²). In this case, the answer to the target question is  NO, x is NOT less than 16 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Bouka2311 · Answer

Could you please explain from where did you get the first formula: (10/2(1+10)/10)

engrossedsid · Answer

instead of this , u can use an easier formula for average of first 10 integers= first term+ last term/2 = 1+10/2 = 11/2 = 5.5

sealberg · Answer

1. Obviously sufficient, as the average of the integers between 1 and 10 is somewhere between 1 and 10 (we don't even need to know exactly where, just that it's less than 16)
2. Obviously insufficient. x could equal 1, 4, 9, 16, 25.. etc.

guenthermat · Answer

The average of the first positive ten integers - but the first positive ten integers of  what ? Why can't the first ten integers be part of a set that, for instance, goes from 100 to 109?

I was aware that (A) would be sufficient if the GMAT refers to 1-10 but I refused this option as the questions doesn't exactly determine the set.

What do u think?

amanvermagmat · Answer

If we are given 'n' numbers in Arithmetic Progression (evenly spaced), there is a formula for their sum which is:

Sum = n/2 (First term + Last term)
Here we are looking at first 10 positive integers: 1, 2, 3, ....10 (these are obviously evenly spaced)

So n=10, first term = 1, last term = 10
Hence sum = (10/2) (1 + 10) = 55

But average = Sum/n = 55/10 = 5.5

amanvermagmat · Answer

Hi

Good query I think. I believe that when a question states 'first 10 positive integers' without giving any other set, 
then it means 'universal first 10 positive integers'.. which means starting from 1 and going upto 10

So first n positive integers = 1,2, 3, 4, ....n
First n non-negative integers = 0, 1, 2, 3, ...

First 4 positive even numbers = 2, 4, 6, 8, 
First two positive multiples of 5 = 5, 10

and so on..

xyz02611 · Answer

What we get from the stem:  X is a positive integer .

Question: Is X less than 16?

1) Integers from 1 to 10 constitute an evenly spaced set (common difference between consecutive terms)

so we can apply the formula:  Average = (First + Last / 2 )

We get X<5.5. Since X is a positive integer, it can be either 5,4,3,2,or 1.
Answer  Yes .

2)  X=100  is not less than 16. Answer is  Yes .

X=4  is less than 16. Answer is  No .

Hence, Answer Choice is A.

bumpbot · Answer

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If x is a positive integer, is x<16?

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If x is a positive integer, is x<16?

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