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# New Set: Number Properties!!!

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25 Mar 2013, 03:50
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The next set of medium/hard DS number properties questions. I'll post OA's with detailed explanations on Friday. Please, post your solutions along with the answers.

1. If x is an integer, what is the value of x?

(1) |23x| is a prime number
(2) $$2\sqrt{x^2}$$ is a prime number.

Solution: new-set-number-properties-149775-40.html#p1205341

2. If a positive integer n has exactly two positive factors what is the value of n?

(1) n/2 is one of the factors of n
(2) The lowest common multiple of n and n + 10 is an even number.

Solution: new-set-number-properties-149775-40.html#p1205355

3. If 0 < x < y and x and y are consecutive perfect squares, what is the remainder when y is divided by x?

(1) Both x and y is have 3 positive factors.
(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers

Solution: new-set-number-properties-149775-60.html#p1205358

4. Each digit of the three-digit integer K is a positive multiple of 4, what is the value of K?

(1) The units digit of K is the least common multiple of the tens and hundreds digit of K
(2) K is NOT a multiple of 3.

Solution: new-set-number-properties-149775-60.html#p1205361

5. If a, b, and c are integers and a < b < c, are a, b, and c consecutive integers?

(1) The median of {a!, b!, c!} is an odd number.
(2) c! is a prime number

Solution: new-set-number-properties-149775-60.html#p1205364

6. Set S consists of more than two integers. Are all the numbers in set S negative?

(1) The product of any three integers in the list is negative
(2) The product of the smallest and largest integers in the list is a prime number.

Solution: new-set-number-properties-149775-60.html#p1205373

7. Is x the square of an integer?

(1) When x is divided by 12 the remainder is 6
(2) When x is divided by 14 the remainder is 2

Solution: new-set-number-properties-149775-60.html#p1205378

8. Set A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the set less than 1/5?

(1) Reciprocal of the median is a prime number
(2) The product of any two terms of the set is a terminating decimal

Solution: new-set-number-properties-149775-60.html#p1205382

9. If [x] denotes the greatest integer less than or equal to x for any number x, is [a] + [b] = 1 ?

(1) ab = 2
(2) 0 < a < b < 2

Solution: new-set-number-properties-149775-60.html#p1205389

10. If N = 3^x*5^y, where x and y are positive integers, and N has 12 positive factors, what is the value of N?

(1) 9 is NOT a factor of N
(2) 125 is a factor of N

Solution: new-set-number-properties-149775-60.html#p1205392

BONUS QUESTION:
11. If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y
(2) x + y - 3 = |1-y|

Solution: new-set-number-properties-149775-60.html#p1205398

Kudos points for each correct solution!!!
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Re: New Set: Number Properties!!! [#permalink]

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11 Apr 2017, 22:26
1
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sindhugclub wrote:
Bunuel wrote:
SOLUTIONS:

1. If x is an integer, what is the value of x?

(1) $$|23x|$$ is a prime number. From this statement it follows that x=1 or x=-1. Not sufficient.

(2) $$2\sqrt{x^2}$$ is a prime number. The same here: x=1 or x=-1. Not sufficient.

(1)+(2) x could be 1 or -1. Not sufficient.

From what I heard from the experts, GMAT's general rule is that square root of a number is always positive on gmat. By this, i can get B as an answer. since root 1 will be 1 and not -1. Please explain.

In addition to what Bunuel said, check out this post: https://www.veritasprep.com/blog/2016/0 ... oots-gmat/
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Re: New Set: Number Properties!!! [#permalink]

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27 Apr 2017, 19:54
Bunuel wrote:
5. If a, b, and c are integers and a < b < c, are a, b, and c consecutive integers?

Note that:
A. The factorial of a negative number is undefined.
B. 0!=1.
C. Only two factorials are odd: 0!=1 and 1!=1.
D. Factorial of a number which is prime is 2!=2.

(1) The median of {a!, b!, c!} is an odd number. This implies that b!=odd. Thus b is 0 or 1. But if b=0, then a is a negative number, so in this case a! is not defined. Therefore a=0 and b=1, so the set is {0!, 1!, c!}={1, 1, c!}. Now, if c=2, then the answer is YES but if c is any other number then the answer is NO. Not sufficient.

(2) c! is a prime number. This implies that c=2. Not sufficient.

(1)+(2) From above we have that a=0, b=1 and c=2, thus the answer to the question is YES. Sufficient.

Hi Bunuel, thanks for putting up such amazing questions.
My Question:
STatement 2: we know that c=2.
Now the question states that a<b<c; (integers) and we know that the factorial of a negative number is not defined.
thus, automatically a=0, b=1.
so shouldnt the answer be B?

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Re: New Set: Number Properties!!! [#permalink]

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27 Apr 2017, 22:00
rachitasetiya wrote:

Hi Bunuel, thanks for putting up such amazing questions.
My Question:
STatement 2: we know that c=2.
Now the question states that a<b<c; (integers) and we know that the factorial of a negative number is not defined.
thus, automatically a=0, b=1.
so shouldnt the answer be B?

Hey rachitasetiya,

The second statement only focuses on the factorial of c. So, we can definitely conclude that c = 2. But there is no information in the second statement, which can help us conclude that a,b and c are consecutive integers.

I guess when you said that " the factorial of a negative number is not defined", you also considered the information given in the first statement, to draw your conclusion.

Please remember, in DS, when we are solving a question using a particular statement we should focus only on the information given in that statement independently.

Hence, from the second statement, we get to know that c = 2 and there is nothing mentioned about a and b. Hence the second statement is not sufficient to answer the question.

It is only by combining, you can conclude that a = 0, b= 1 and c =2.

I hope the above explanation is clear.

Thanks,
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Re: New Set: Number Properties!!! [#permalink]

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23 Aug 2017, 08:03
Bunuel wrote:
7. Is x the square of an integer?

The question basically asks whether x is a perfect square (a perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square).

Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

(1) When x is divided by 12 the remainder is 6. Given that $$x=12q+6=6(2q+1)=2*3*(2q+1)$$. Now, since 2q+1 is an odd number then the power of 2 in x will be odd (1), thus x cannot be a perfect square. Sufficient.

(2) When x is divided by 14 the remainder is 2. Given that $$x=14p+2$$. So, x could be 2, 16, 30, ... Thus, x may or may not be a perfect square. Not sufficient.

hi

2 * 3 * (2q + 1)
obviously, (2q + 1) is an odd number, however, since (2q + 1) is an odd number, the power of 2 in x will be odd...
maybe it is very obvious to you, but I am totally stumped here... please help me understand it ...

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Re: New Set: Number Properties!!! [#permalink]

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23 Aug 2017, 10:26
gmatcracker2017 wrote:
Bunuel wrote:
7. Is x the square of an integer?

The question basically asks whether x is a perfect square (a perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square).

Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

(1) When x is divided by 12 the remainder is 6. Given that $$x=12q+6=6(2q+1)=2*3*(2q+1)$$. Now, since 2q+1 is an odd number then the power of 2 in x will be odd (1), thus x cannot be a perfect square. Sufficient.

(2) When x is divided by 14 the remainder is 2. Given that $$x=14p+2$$. So, x could be 2, 16, 30, ... Thus, x may or may not be a perfect square. Not sufficient.

hi

2 * 3 * (2q + 1)
obviously, (2q + 1) is an odd number, however, since (2q + 1) is an odd number, the power of 2 in x will be odd...
maybe it is very obvious to you, but I am totally stumped here... please help me understand it ...

x=2*3*odd. Since an odd number does not have 2 in it, then 2 in x (highlighted) will be the only 2 in x. Since a perfect square always has even powers of its prime factors and 2 is in odd power (1), then x cannot be a perfect square.
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Re: New Set: Number Properties!!! [#permalink]

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24 Aug 2017, 07:54
Bunuel wrote:
gmatcracker2017 wrote:
Bunuel wrote:
7. Is x the square of an integer?

The question basically asks whether x is a perfect square (a perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square).

Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

(1) When x is divided by 12 the remainder is 6. Given that $$x=12q+6=6(2q+1)=2*3*(2q+1)$$. Now, since 2q+1 is an odd number then the power of 2 in x will be odd (1), thus x cannot be a perfect square. Sufficient.

(2) When x is divided by 14 the remainder is 2. Given that $$x=14p+2$$. So, x could be 2, 16, 30, ... Thus, x may or may not be a perfect square. Not sufficient.

hi

2 * 3 * (2q + 1)
obviously, (2q + 1) is an odd number, however, since (2q + 1) is an odd number, the power of 2 in x will be odd...
maybe it is very obvious to you, but I am totally stumped here... please help me understand it ...

x=2*3*odd. Since an odd number does not have 2 in it, then 2 in x (highlighted) will be the only 2 in x. Since a perfect square always has even powers of its prime factors and 2 is in odd power (1), then x cannot be a perfect square.

hi man

Its now crystal clear to me...thanks bro

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Re: New Set: Number Properties!!! [#permalink]

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25 Aug 2017, 18:25
Bunuel wrote:
2. If a positive integer n has exactly two positive factors what is the value of n?

Notice that, n has exactly two positive factors simply means that n is a prime number, so its factors are 1 and n itself.

(1) n/2 is one of the factors of n. Since n/2 cannot equal to n, then n/2=1, thus n=2. Sufficient.

(2) The lowest common multiple of n and n + 10 is an even number. If n is an odd prime, then n+10 is also odd. The LCM of two odd numbers cannot be even, therefore n is an even prime, so 2. Sufficient.

hi man

can this problem be seen as under ....

n / n / 2 = integer

or n * 2/n = integer
or 2n/n = integer

now, whichever prime "n" can take, the equation will hold true....

under circumstance as such, statement 1 is not sufficient .....

please say to me what is wrong here...

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Re: New Set: Number Properties!!! [#permalink]

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26 Aug 2017, 01:19
gmatcracker2017 wrote:
Bunuel wrote:
2. If a positive integer n has exactly two positive factors what is the value of n?

Notice that, n has exactly two positive factors simply means that n is a prime number, so its factors are 1 and n itself.

(1) n/2 is one of the factors of n. Since n/2 cannot equal to n, then n/2=1, thus n=2. Sufficient.

(2) The lowest common multiple of n and n + 10 is an even number. If n is an odd prime, then n+10 is also odd. The LCM of two odd numbers cannot be even, therefore n is an even prime, so 2. Sufficient.

hi man

can this problem be seen as under ....

n / n / 2 = integer

or n * 2/n = integer
or 2n/n = integer

now, whichever prime "n" can take, the equation will hold true....

under circumstance as such, statement 1 is not sufficient .....

please say to me what is wrong here...

n divided by n/2 gives 2 irrespective of n, so the way you are doing cannot help to get sufficiency or insufficiency of the statement.
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Re: New Set: Number Properties!!! [#permalink]

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26 Aug 2017, 06:10
Bunuel wrote:
gmatcracker2017 wrote:
Bunuel wrote:
2. If a positive integer n has exactly two positive factors what is the value of n?

Notice that, n has exactly two positive factors simply means that n is a prime number, so its factors are 1 and n itself.

(1) n/2 is one of the factors of n. Since n/2 cannot equal to n, then n/2=1, thus n=2. Sufficient.

(2) The lowest common multiple of n and n + 10 is an even number. If n is an odd prime, then n+10 is also odd. The LCM of two odd numbers cannot be even, therefore n is an even prime, so 2. Sufficient.

hi man

can this problem be seen as under ....

n / n / 2 = integer

or n * 2/n = integer
or 2n/n = integer

now, whichever prime "n" can take, the equation will hold true....

under circumstance as such, statement 1 is not sufficient .....

please say to me what is wrong here...

n divided by n/2 gives 2 irrespective of n, so the way you are doing cannot help to get sufficiency or insufficiency of the statement.

okay man ...

thanks ....

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Re: New Set: Number Properties!!! [#permalink]

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27 Aug 2017, 04:39
Bunuel wrote:
11. If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel

Can you please elaborate the solution of this one?
I did it with critical point method and ended with 2 equations in case of x>2(x+y=4) and x<2(x+y=0) and 2 equations with x<1(x+2y=4) and x>1(x=2).
How to proceed further?

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Re: New Set: Number Properties!!! [#permalink]

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27 Aug 2017, 05:17
nishantt7 wrote:
Bunuel wrote:
11. If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel

Can you please elaborate the solution of this one?
I did it with critical point method and ended with 2 equations in case of x>2(x+y=4) and x<2(x+y=0) and 2 equations with x<1(x+2y=4) and x>1(x=2).
How to proceed further?

For (1): In $$|x - 2| \lt 2 - y$$, the left hand side of the inequality (|x - 2|) is an absolute value. An absolute value of a number cannot be negative, so the left hand side of the inequality (|x - 2|) is not negative, which means that the right hand side of the inequality, which is greater than the left hand side, also cannot be negative. This gives $$0 \lt 2 - y$$.

For (2): We know from the stem that y is a positive integer. Since y is a positive integer (1, 2, 3, ...), then 1 - y = 1 - positive = non-positive, which measn that |1-y|=-(1-y). Therefore, for (2) we'd have $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hope it's clear.
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Re: New Set: Number Properties!!! [#permalink]

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25 Sep 2017, 09:59
Bunuel wrote:
9. If [x] denotes the greatest integer less than or equal to x for any number x, is [a] + [b] = 1 ?

Given that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

(1) ab = 2. First of all this means that a and b are of the same sign.

If both are negative, then the maximum value of [a] + [b] is -2, for any negative a and b. So, this case is out.

If both are positive, then in order [a] + [b] = 1 to hold true, must be true that [a]=0 and [b]=1 (or vise-versa). Which means that $$0\leq{a}<1$$ and $$1\leq{b}<2$$ (or vise-versa). But in this case ab cannot be equal to 2. So, this case is also out.

We have that the answer to the question is NO. Sufficient.

(2) 0 < a < b < 2. If a=1/2 and b=1, then [a] + [b] = 0 + 1 = 1 but if a=1/4 and b=1/2, then [a] + [b] = 0 + 0 = 0. Not sufficient.

Can someone explain the reasoning behind "If both are negative, then the maximum value of [a] + [b] is -2, for any negative a and b"?
how did you arrive at this maximum value?

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Re: New Set: Number Properties!!! [#permalink]

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25 Sep 2017, 16:47
1
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Expert's post
Ace800 wrote:
Bunuel wrote:
9. If [x] denotes the greatest integer less than or equal to x for any number x, is [a] + [b] = 1 ?

Given that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

(1) ab = 2. First of all this means that a and b are of the same sign.

If both are negative, then the maximum value of [a] + [b] is -2, for any negative a and b. So, this case is out.

If both are positive, then in order [a] + [b] = 1 to hold true, must be true that [a]=0 and [b]=1 (or vise-versa). Which means that $$0\leq{a}<1$$ and $$1\leq{b}<2$$ (or vise-versa). But in this case ab cannot be equal to 2. So, this case is also out.

We have that the answer to the question is NO. Sufficient.

(2) 0 < a < b < 2. If a=1/2 and b=1, then [a] + [b] = 0 + 1 = 1 but if a=1/4 and b=1/2, then [a] + [b] = 0 + 0 = 0. Not sufficient.

Can someone explain the reasoning behind "If both are negative, then the maximum value of [a] + [b] is -2, for any negative a and b"?
how did you arrive at this maximum value?

Hi
To get the max value, take a and b as the smallest possible negative values..
So a =b= -0.00000000000....1
Least INTEGER for both a and b will be -1..
Thus [a]+[b]=-1+(-1)=-2
So -2 is the max value of a and b are NEGATIVE numbers.

Hope it helps
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Re: New Set: Number Properties!!! [#permalink]

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25 Sep 2017, 22:47
Oh ok. Thanks, Chetan!

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27 Dec 2017, 15:39
VeritasPrepKarishma wrote:
sindhugclub wrote:
Bunuel wrote:
SOLUTIONS:

1. If x is an integer, what is the value of x?

(1) $$|23x|$$ is a prime number. From this statement it follows that x=1 or x=-1. Not sufficient.

(2) $$2\sqrt{x^2}$$ is a prime number. The same here: x=1 or x=-1. Not sufficient.

(1)+(2) x could be 1 or -1. Not sufficient.

From what I heard from the experts, GMAT's general rule is that square root of a number is always positive on gmat. By this, i can get B as an answer. since root 1 will be 1 and not -1. Please explain.

In addition to what Bunuel said, check out this post: https://www.veritasprep.com/blog/2016/0 ... oots-gmat/

Hi
Could you explain why this would be wrong here?

statement 2 tells us $$2 \sqrt{x^2}$$ is prime number

the expression $$2 * [x^2 ] ^ \frac{1}{2}$$ becomes $$2 x$$[ square canceling out with the square root ]

if the expression summarises to $$2x$$then statement 2 becomes$$2x$$ is a prime number and $$x =1$$is the only value which satisfies.

If this is wrong, could you explain the why? It will be really helpful.
Thankyou

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Re: New Set: Number Properties!!! [#permalink]

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27 Dec 2017, 20:13
mtk10 wrote:
VeritasPrepKarishma wrote:
sindhugclub wrote:

From what I heard from the experts, GMAT's general rule is that square root of a number is always positive on gmat. By this, i can get B as an answer. since root 1 will be 1 and not -1. Please explain.

In addition to what Bunuel said, check out this post: https://www.veritasprep.com/blog/2016/0 ... oots-gmat/

Hi
Could you explain why this would be wrong here?

statement 2 tells us $$2 \sqrt{x^2}$$ is prime number

the expression $$2 * [x^2 ] ^ \frac{1}{2}$$ becomes $$2 x$$[ square canceling out with the square root ]

if the expression summarises to $$2x$$then statement 2 becomes$$2x$$ is a prime number and $$x =1$$is the only value which satisfies.

If this is wrong, could you explain the why? It will be really helpful.
Thankyou

You could see that this is wrong by considering the values given in the solution: both x=1 or x=-1 make $$2\sqrt{x^2}$$ prime number.

That's because $$\sqrt{x^2}=|x|$$ NOT x.

The point here is that since square root function cannot give negative result then $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$. That is why $$\sqrt{x^2}=|x|$$.

Hope it's clear.
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Re: New Set: Number Properties!!! [#permalink]

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12 Jan 2018, 01:05
Bunuel wrote:
8. Set A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the set less than 1/5?

(1) Reciprocal of the median is a prime number. If all the terms equal 1/2, then the median=1/2 and the answer is NO but if all the terms equal 1/7, then the median=1/7 and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of 1/2 or/and 1/5. Thus the median could be 1/2, 1/5 or (1/5+1/2)/2=7/20. None of the possible values is less than 1/5. Sufficient.

Theory:
Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be the terminating decimal.

We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.

Questions testing this concept:
http://gmatclub.com/forum/does-the-deci ... 89566.html
http://gmatclub.com/forum/any-decimal-t ... 01964.html
http://gmatclub.com/forum/if-a-b-c-d-an ... 25789.html
http://gmatclub.com/forum/700-question-94641.html
http://gmatclub.com/forum/is-r-s2-is-a- ... 91360.html
http://gmatclub.com/forum/pl-explain-89566.html
http://gmatclub.com/forum/which-of-the- ... 88937.html

Hi @Bunuel/ chetan2u,

The question is essentially asking whether "median of the set is less than 1/5?". So if we have median EQUALto 1/5 the answer CAN'T be B.

Please elucidate: what am I missing here? IMHO the answer should be "E".

Regards
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Re: New Set: Number Properties!!! [#permalink]

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12 Jan 2018, 02:42
gmatexam439 wrote:
Bunuel wrote:
8. Set A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the set less than 1/5?

(1) Reciprocal of the median is a prime number. If all the terms equal 1/2, then the median=1/2 and the answer is NO but if all the terms equal 1/7, then the median=1/7 and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of 1/2 or/and 1/5. Thus the median could be 1/2, 1/5 or (1/5+1/2)/2=7/20. None of the possible values is less than 1/5. Sufficient.

Theory:
Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be the terminating decimal.

We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.

Questions testing this concept:
http://gmatclub.com/forum/does-the-deci ... 89566.html
http://gmatclub.com/forum/any-decimal-t ... 01964.html
http://gmatclub.com/forum/if-a-b-c-d-an ... 25789.html
http://gmatclub.com/forum/700-question-94641.html
http://gmatclub.com/forum/is-r-s2-is-a- ... 91360.html
http://gmatclub.com/forum/pl-explain-89566.html
http://gmatclub.com/forum/which-of-the- ... 88937.html

Hi @Bunuel/ chetan2u,

The question is essentially asking whether "median of the set is less than 1/5?". So if we have median EQUALto 1/5 the answer CAN'T be B.

Please elucidate: what am I missing here? IMHO the answer should be "E".

Regards

Hi..
If you ask Q is 1/5 less than 1/5?... Ans will be NO
If you ask .. is 1/2 less than 1/5 .... Ans is again NO
So evwrytime and is NO..
Sufficient

But say the Q was - is x >1/5?
If 1/2 .Ans will be Yes..
If 1/5.. Ans will be NO
So insufficient...
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: New Set: Number Properties!!! [#permalink]

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12 Jan 2018, 02:51
gmatexam439 wrote:
Bunuel wrote:
8. Set A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the set less than 1/5?

(1) Reciprocal of the median is a prime number. If all the terms equal 1/2, then the median=1/2 and the answer is NO but if all the terms equal 1/7, then the median=1/7 and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of 1/2 or/and 1/5. Thus the median could be 1/2, 1/5 or (1/5+1/2)/2=7/20. None of the possible values is less than 1/5. Sufficient.

Theory:
Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be the terminating decimal.

We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.

Questions testing this concept:
http://gmatclub.com/forum/does-the-deci ... 89566.html
http://gmatclub.com/forum/any-decimal-t ... 01964.html
http://gmatclub.com/forum/if-a-b-c-d-an ... 25789.html
http://gmatclub.com/forum/700-question-94641.html
http://gmatclub.com/forum/is-r-s2-is-a- ... 91360.html
http://gmatclub.com/forum/pl-explain-89566.html
http://gmatclub.com/forum/which-of-the- ... 88937.html

Hi @Bunuel/ chetan2u,

The question is essentially asking whether "median of the set is less than 1/5?". So if we have median EQUALto 1/5 the answer CAN'T be B.

Please elucidate: what am I missing here? IMHO the answer should be "E".

Regards

For (2) all possible cases (including the median of 1/5) give a NO answer to the question. A definite No answer to the question means that the statement is sufficient.
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Re: New Set: Number Properties!!! [#permalink]

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13 Jan 2018, 01:59
Thank you Bunuel and Chetan. I committed a silly mistake here.

Regards
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Re: New Set: Number Properties!!!   [#permalink] 13 Jan 2018, 01:59

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# New Set: Number Properties!!!

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