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If the remainder is 1, when integer n(n>1) is divided by 3,

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If the remainder is 1, when integer n(n>1) is divided by 3, [#permalink] New post 30 Jul 2020, 15:20
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n is an integer greater than 1. If the remainder is 1, when n is divided by 3, then (n² + n - 2) must be divisible by which double-digit number?

A. 12
B. 14
C. 16
D. 18
E. 20
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Re: If the remainder is 1, when integer n(n>1) is divided by 3, [#permalink] New post 30 Jul 2020, 22:59
When n is divided by 3, the remainder is 1.
So lets say n is 7, when when divided by 3 gives remainder as 1.
Now, according to the equation, (n^2+n-2)
49 + 7 - 2 = 54
Only 18 can divide 54 completely. Thus, answer is D :)
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Re: If the remainder is 1, when integer n(n>1) is divided by 3, [#permalink] New post 03 Aug 2020, 04:45
In general if we have integers x,y,m,and r such that

x/y=m+(r/y) then r is the remainder and m is the whole number part of the division
e.g. 4/3=1+1/3 here x=4, y=3, m=1, r=1


Let n and m be integers

n/3=m+(1/3)

m is the whole number part of the division and 1 in (1/3) represents the remainder.

multiple 3 on both sides

n=3m+1

therefore

n^2-n-2= (n+2)(n-1)= ((3m+1)+2)((3m+1)-1) =(3m+3)(3m) =3(m+1)3m= 9(m+1)(m)

Finally since m and m+1 are consecutive integers one of them must be even therefore we can factor out a 2 from it giving us

=9*2*((m+1)m)/2
=18*((m+1)(m))/2

Since 18 is a factor of n^2-n-2. Then n^2-n-2 is divisible by 18.

Final Answer: D
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Re: If the remainder is 1, when integer n(n>1) is divided by 3, [#permalink] New post 04 Aug 2020, 12:23
Saraforgre wrote:
n is an integer greater than 1. If the remainder is 1, when n is divided by 3, then (n² + n - 2) must be divisible by which double-digit number?

A. 12
B. 14
C. 16
D. 18
E. 20


If n/3 yields a remainder of 1, then n must be one greater than a multiple of 3. Since the problem specifies that the given equation MUST be divisible by one of the answers choices, we know that we can choose any value that satisfies the criteria.

Let n = 4. (4)^2 + 4 - 2 = 18, so D is clearly the correct answer.
Re: If the remainder is 1, when integer n(n>1) is divided by 3,   [#permalink] 04 Aug 2020, 12:23
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If the remainder is 1, when integer n(n>1) is divided by 3,

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