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Around the World in 80 Questions (Day 9): If n is a positive integer

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Bunuel
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Around the World in 80 Questions (Day 9): If n is a positive integer [#permalink] Around the World in 80 Questions (Day 9): If n is a positive integer   27 Jul 2023, 08:00
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If n is a positive integer, what is the value of (x - 3)^n + (3 - x)^n ?

(1) n divided by 6 gives the remainder of 1.
(2) 21 divided by n gives the same remainder as 20 divided by n.


 


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nivivacious
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Re: Around the World in 80 Questions (Day 9): If n is a positive integer [#permalink] Around the World in 80 Questions (Day 9): If n is a positive integer   27 Jul 2023, 08:17
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Bunuel wrote:
If n is a positive integer, what is the value of (x - 3)^n + (3 - x)^n ?

(1) n divided by 6 gives the remainder of 1.
(2) 21 divided by n gives the same remainder as 20 divided by n.


 


This question was provided by GMAT Club
for the Around the World in 80 Questions

Win over $20,000 in prizes: Courses, Tests & more

 



(1) n divided by 6 gives the remainder of 1.
Therefore n is in the form 6k+1, which is an odd number.
So value of (x - 3)^n + (3 - x)^n is 0.
Hence sufficient

(2) 21 divided by n gives the same remainder as 20 divided by n.
This means n must be a factor of 1, because the difference between 21 and 20 is 1. Therefore, n can only be 1.
if n is 1, it is odd and the value of (x - 3)^n + (3 - x)^n is 0.
Hence sufficient

Hence D
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Re: Around the World in 80 Questions (Day 9): If n is a positive integer [#permalink] Around the World in 80 Questions (Day 9): If n is a positive integer   27 Jul 2023, 08:36
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answer is D.

STATEMENT 1 : n divided by 6 gives the remainder of 1.
so n is odd. (3-x)^n = ((-1)*(x-3))^n = (-1)^n * (x-3)^n = -((x-3)^n) ------ since n is odd
therefore the required value is 0. this statment is sufficient

STATEMENT 2 : 21 divided by n gives the same remainder as 20 divided by n.
21 and 20 are co prime. and n is positive. thus n is 1. placing 1 in the equation, we can determine the value to be 0. this statment is sufficient

EACH ALONE IS SUFFICIENT. SO ANSWER IS D
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Re: Around the World in 80 Questions (Day 9): If n is a positive integer [#permalink] Around the World in 80 Questions (Day 9): If n is a positive integer   27 Jul 2023, 08:42
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Bunuel wrote:
If n is a positive integer, what is the value of (x - 3)^n + (3 - x)^n ?

(1) n divided by 6 gives the remainder of 1.
(2) 21 divided by n gives the same remainder as 20 divided by n.


 


This question was provided by GMAT Club
for the Around the World in 80 Questions

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(x - 3)^n + (3 - x)^n

(x - 3)^n + (-1)^n(x - 3)^n

(1) n divided by 6 gives the remainder of 1.

n is odd, hence (1)^n = -1

(x - 3)^n - (x - 3)^n = 0

The value is zero. Sufficient.

(2) 21 divided by n gives the same remainder as 20 divided by n.

20 and 21 are consecutive numbers, hence co-primes. Only common factor = 1

n is odd, hence (1)^n = -1

(x - 3)^n - (x - 3)^n = 0

Sufficient.

IMO D
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Re: Around the World in 80 Questions (Day 9): If n is a positive integer [#permalink]
27 Jul 2023, 08:42
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Bunuel
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