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[swapnil_swapnil]

If x and n are positive integers, is n = 1?

(1) The sum of n consecutive integers, starting at x, is divisible by xn.
(2) The product of n consecutive integers, starting at x, is divisible by x^n.

reason provided to rule out 1 is:
'.....
Statement (1) asserts that the sum of n consecutive integers, starting at x, is divisible by xn. First, we can connect this with other facts about consecutive integers. It turns out that the sum of n consecutive integers is divisible by n if and only if n is odd. (The reason is that the average number in a set of consecutive integers is actually the middle integer if you have 3, 5, or some other odd number of integers. But if you have an even number of integers, there is no middle integer, and the average number is not an integer. This matters because the sum of n consecutive integers divided by n IS the average number in a set of consecutive integers.) So we rule out even values of n. However, if we simply let x be 1, then the condition is satisfied by n = 1, 3, 5, or any other positive odd integer. We do not know whether n is equal to 1. INSUFFICIENT.

the explanation talks about divisibilty by 'n"....but the questions divisibility by "xn"

for example: if x = 7 and n = 3 then
sum 7+8+9 will be divisible by 3 .

but the quetions asks for divisibility by 7.3 (x.n)
....'

[RonPurewal]

If x and n are positive integers, is n = 1?

(1) The sum of n consecutive integers, starting at x, is divisible by xn.
(2) The product of n consecutive integers, starting at x, is divisible by x^n.

reason provided to rule out 1 is:
'.....
Statement (1) asserts that the sum of n consecutive integers, starting at x, is divisible by xn. First, we can connect this with other facts about consecutive integers. It turns out that the sum of n consecutive integers is divisible by n if and only if n is odd. (The reason is that the average number in a set of consecutive integers is actually the middle integer if you have 3, 5, or some other odd number of integers. But if you have an even number of integers, there is no middle integer, and the average number is not an integer. This matters because the sum of n consecutive integers divided by n IS the average number in a set of consecutive integers.) So we rule out even values of n. However, if we simply let x be 1, then the condition is satisfied by n = 1, 3, 5, or any other positive odd integer. We do not know whether n is equal to 1. INSUFFICIENT.

the explanation talks about divisibilty by 'n"....but the questions divisibility by "xn"

for example: if x = 7 and n = 3 then
sum 7+8+9 will be divisible by 3 .

but the quetions asks for divisibility by 7.3 (x.n)
....'

hi -

the explanation FIRST discusses divisibility by n, because that discussion is needed to rephrase the given information. specifically, if the sum of n integers is divisible by n, then n must be odd.

the example then DOES go on to discuss "x", although the only value of x that's considered is x = 1.
however, this is the only value of x that needs to be considered. the question asks only for n, so we only have to consider those values of x that are NECESSARY to help us answer the question.
by just using x = 1, and thereby reducing the question of divisibility by xn to the simpler issue of divisibility by n, we can already tell that the statement is insufficient. therefore, there's no reason to consider other values of x.