9.If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r ?
(1) n is not divisible by 2.
(2) n is not divisible by 3.
I know this is not new to the forum, but I couldnt find an explanation that was clear to me. Can someone provide detailed reasoning?
For statement 1. take n = 5 the number becomes 4*6 when divided by 24 gives remainder 0
take n = 7 the number becomes 6*8 when divided by 24 gives remainder 0
take n = 9 the number becomes 8*10 when divided by 24 gives remainder different than 0. Thus not sufficient.
For statement 2. take n= 5 and n=7 it will give remainder as 0
if you take n =
14 number becomes 13*15 when divided by 24 gives remainder other than 0. Thus not sufficient.
So only answers left are either C or E
Now you must have observed that when n is not multiple of 2 or not multiple of 3 independently we are unable to answer the question. But when n is neither multiple of 2 nor 3 then we are able to answer the question as r=0 in all those cases.
Another way of looking at it is.... take n-1 , n , n+1 these are 3 consecutive integers, we know we will have both the multiple of 2 and 3 in three consecutive integer. If that integer is same i.e. n is multiple of both 2 and 3 , then n-1 and n+1 will never have any factor for 24.
But if n is not divisible by both 2 and 3 that means n-1 and n+1 will be both be even with one of them having 3 as factor. One another point to be noted is , since n-1 and n+1 are consecutive even numbers that implies one of them must be divisible by 4.
so (n-1)(n+1) will be always divisible by 2*4*3 = 24 hence r will always remain as 0
(When n is neither divisible by 2 nor by 3)
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