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tough DS - Special operations

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Source: — Data Sufficiency |

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by Rahul@gurome » Mon Nov 01, 2010 5:25 am
replayyyy wrote:If for any positive integer x , d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

1. D[x] - d[x] = 0
2. D[3x] = 3
Smallest odd divisor of any positive integer is 1. Thus, d[x] = 1 for any integer x > 0

Statement 1: D[x] - d[x] = 0 => D[x] = d[x] = 1
Now, there may be 2 cases:
  • (1) x = 1 => x odd
    (2) x = 1*(Any even integer) => x even.
Not sufficient.

Statement 2: D[3x] = 3 => Largest odd divisor of 3x is 3
Now, there may be 2 cases:
  • (1) 3x = 3*1 => x = 1 => x odd
    (2) 3x = 3*(Any even integer) => x even.
Not sufficient.

The correct answer is E.

Note: This problem is solved assuming that 1 is a divisor of any number. To avoid ambiguity 1, -1, n and -n are termed as Trivial Divisors of n, whereas other divisors are termed as Non-trivial Divisors of n.
Last edited by Rahul@gurome on Mon Nov 01, 2010 8:39 am, edited 1 time in total.
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by replayyyy » Mon Nov 01, 2010 8:14 am
Rahul@gurome wrote:
replayyyy wrote:If for any positive integer x , d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

1. D[x] - d[x] = 0
2. D[3x] = 3
Smallest odd divisor of any positive integer is 1. Thus, d[x] = 1 for any integer x > 0

Statement 1: D[x] - d[x] = 0 => D[x] = d[x] = 1 => x = 1

Sufficient.

Statement 2: D[3x] = 3 => Largest odd divisor of 3x is 3
Now, there may be 2 cases:
  • (1) 3x = 3*1 => x = 1 => x odd
    (2) 3x = 3*(Any even integer) => x even.
Not sufficient.

The correct answer is A.

Note: This problem is solved assuming that 1 is a divisor of any number. To avoid ambiguity 1, -1, n and -n are termed as Trivial Divisors of n, whereas other divisors are termed as Non-trivial Divisors of n.
The official answer is E. If we know that the greatest and the smallest odd divisors are equal, this means that they are = 1, but x does not have to be 1, it could be 2,4,8,16 or any power of 2. I had doubts about why is E because I thought it is A too, when posting the question, but I`ve later realizied the possibility of x=1 or x=2 or x=4 ...
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by Rahul@gurome » Mon Nov 01, 2010 8:40 am
replayyyy wrote:...
The official answer is E. If we know that the greatest and the smallest odd divisors are equal, this means that they are = 1, but x does not have to be 1, it could be 2,4,8,16 or any power of 2. I had doubts about why is E because I thought it is A too, when posting the question, but I`ve later realizied the possibility of x=1 or x=2 or x=4 ...
Thanks for pointing out the mistake.
Edited the reply.
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by GmatKiss » Mon Aug 15, 2011 4:51 am
Is it not D?

1) D[x] - d[x] = 0 means, D[x] and d[x] are same, which applies to only 1. Sufficient
2) D[3x] = 3 means, Largest divisor of 3x is 3, 3 is the largest divisor only for 3!, so x should be 1. Sufficient
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by codytravers » Mon Aug 15, 2011 12:40 pm
I was onboard with your answer at first, but think about this:

1) For 1, you can also pick 2, and D[x]-d[x] would = 0 (1-1)
2) For 2, you can also pick 2 as x (Largest odd divisor of 2 is 1, which would result in 3)

E
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by GmatKiss » Tue Aug 16, 2011 1:32 am
oh yes!! thanks a lot :)
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