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Math Expert V
Joined: 02 Sep 2009
Posts: 57287
Let b and x be positive integers. If b is the greatest divisor of x th  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 24% (02:53) correct 76% (02:36) wrong based on 55 sessions

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Let b and x be positive integers. If b is the greatest divisor of x that is less than x, is the sum of the divisors of x, which are less than x itself and greater than one, greater than 2b?

(1) b^2 = x
(2) 2b = x

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Retired Moderator D
Joined: 25 Feb 2013
Posts: 1192
Location: India
GPA: 3.82
Re: Let b and x be positive integers. If b is the greatest divisor of x th  [#permalink]

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1
Bunuel wrote:
Let b and x be positive integers. If b is the greatest divisor of x that is less than x, is the sum of the divisors of x, which are less than x itself and greater than one, greater than 2b?

(1) b^2 = x
(2) 2b = x

Statement 1: $$b$$ is the greatest divisor of $$x$$ and $$x=b^2$$, so $$b$$ must be prime.

For eg. $$b=7$$, then $$x=49$$. Divisors of $$x=1,7,49$$. Greatest divisor $$7=b$$

$$b$$ cannot be a composite number. for eg. if $$b=4$$, then $$x=16$$. Divisors of $$x= 1,2,4,8,16$$. Greatest divisor $$8$$ which is not equal to $$b$$.

So if $$b =p_1$$, a prime number and as $$x$$ is square of a prime number, $$x={p_1}^2$$, divisors of $$x$$ will be $$1$$, $$p_1$$ & $${p_1}^2$$ only.

Now the divisors of $$x$$, which are less than $$x$$ itself and greater than one is $$p_1=b$$ which is not greater than $$2b=2p_1$$. Sufficient

Statement 2: if $$b=3$$, $$x=6$$. Divisors of $$x= 1,2,3,6$$. sum of divisors of $$x$$, which are less than $$x$$ itself and greater than one is $$2+3=5<2*3$$

if $$b=6$$, $$x=12$$. Divisors of $$x=1,2,3,4,6,12$$. sum of divisors of $$x$$, which are less than $$x$$ itself and greater than one is $$2+3+4+6=15>2*6$$. Hence Insufficient

Option A
Intern  Joined: 28 Dec 2016
Posts: 7
Re: Let b and x be positive integers. If b is the greatest divisor of x th  [#permalink]

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Ans is C.
Considering question stem, we have b as greatest divisor of x. For any no, it's greatest divisor will be half or less than half of itself, for it being a non prime.
Consider st1.
B^2=x

5x5 =25, sum of divisors = 5 lesser than 5.
6×6=36,sum of divisors = 2+3+4+6+9+18 greater than 6.cant answer. Not sufficient

Consider st2
2b=x
Eg
2x4=8
Sum of divisors of 8 = 2+4 less than 8
2×6=12
Sum of divisors of 12 = 2+3 less than 12.
2x8=16
Sum of divisors of 16 = 2+4+8 less than 16
2x9=18
Sum of divisors of 18 =2+3+6+9 greater than 18.
Not sufficient.

Considering both st1 n st2

b^2=2b gives b=2
For b=2 sum of divisors of 2b is 2 which is less that 2b=4. Hence sufficient.

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Senior Manager  S
Joined: 15 Jan 2017
Posts: 345
Re: Let b and x be positive integers. If b is the greatest divisor of x th  [#permalink]

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I picked D: Why is it A?
St 1 - x^2 = b -->x = b * b
Thus,x > 2b > b (as asked by the question. I think I have made a mistake here in the interpretation of the question). Thus x is the largest value here.

St 2 --> 2b = x--> 2b/b = 2. So, one factor is 2. b = x/2. What I got was --> x > b and x = 2b. So I am not sure why its not suff
Retired Moderator D
Joined: 25 Feb 2013
Posts: 1192
Location: India
GPA: 3.82
Let b and x be positive integers. If b is the greatest divisor of x th  [#permalink]

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Madhavi1990 wrote:
I picked D: Why is it A?
St 1 - x^2 = b -->x = b * b
Thus,x > 2b > b (as asked by the question. I think I have made a mistake here in the interpretation of the question). Thus x is the largest value here.

St 2 --> 2b = x--> 2b/b = 2. So, one factor is 2. b = x/2. What I got was --> x > b and x = 2b. So I am not sure why its not suff

Hi Madhavi1990

I think you interpreted the question wrongly. we need to calculate sum of divisors of x but you are focusing only on the greatest divisor of x i.e b. for statement 2 if b=6, then x=12 and it will have other factors as well apart from 2. Let b and x be positive integers. If b is the greatest divisor of x th   [#permalink] 18 Dec 2017, 12:54
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