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17 Jun 2009, 22:07
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Can a sequence of different positive integers \(X_{1}, X_{2}, X_{3}, . . . , X_{n}\) be ordered in such a way that the difference between any two consecutive integers is less than 2?
(1) Each term of the sequence has exactly 4 factors.
(2) \(X_{1}*X_{2}*X_{3}*...*X_{n-1}*X_{n}\) is divisible by \(2^{n}\)
(1) Each term of the sequence has exactly 4 factors.
(2) \(X_{1}*X_{2}*X_{3}*...*X_{n-1}*X_{n}\) is divisible by \(2^{n}\)
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17 Jun 2009, 23:04
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In the highlighted part above, did you mean by any two successive integers ?
If you say any consqcutive integers, then it is obviously yes but neither statement is required and D. If you meant "any two successive integers", then it is A.
(1) is suff. If each term of the sequence has exactly 4 factors, then obviously the difference between any two succesive integers is not <2.
(2) is not suff. To have the difference of < 2 (i.e. 1) between two succesive integers, the integers have to be consecutive.
* If n is even, then the difference could be 2 or >2. If n = 2, and x1 = 2 and x2 = 6, x1+x2 = 8, which is divisible nby 2^n.
* If n is odd, then the difference could be <2. If n = 3, x1 = 7, x2 =8 and x3 = 9, then x1+x2+x3 = 24, which is divisible by 2^n. NSF.
It is A.
18 Jun 2009, 08:53
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The consecutive word in the Q refers to numbers in the sequence, not in general
What is the answer?
X_{1}*X_{2}*X_{3}*...*X_{n-1}*X_{n} is divisible by 2^{n}
If you take 6,7,8 it is divisible by 2^3
The way this Q is worded is confusing to me.
It asks can there be such a sequence whose successive terms have a difference less than 2. So Doesnt that mean if we can find one such sequence, we have sufficiency? Or Am I missing some thing basic here? If we have suff in 2 doesnt that contradict with A which yields no
What is the answer?
X_{1}*X_{2}*X_{3}*...*X_{n-1}*X_{n} is divisible by 2^{n}
If you take 6,7,8 it is divisible by 2^3
The way this Q is worded is confusing to me.
It asks can there be such a sequence whose successive terms have a difference less than 2. So Doesnt that mean if we can find one such sequence, we have sufficiency? Or Am I missing some thing basic here? If we have suff in 2 doesnt that contradict with A which yields no
