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
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Rahul@gurome
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Smallest odd divisor of any positive integer is 1. Thus, d[x] = 1 for any integer x > 0replayyyy 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
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.
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.
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.
Rahul Lakhani
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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 ...Rahul@gurome wrote:Smallest odd divisor of any positive integer is 1. Thus, d[x] = 1 for any integer x > 0replayyyy 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
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:Not sufficient.
- (1) 3x = 3*1 => x = 1 => x odd
(2) 3x = 3*(Any even integer) => x even.
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.
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Rahul@gurome
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Thanks for pointing out the mistake.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 ...
Edited the reply.
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GmatKiss
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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
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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codytravers
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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
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
