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For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/

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For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/  [#permalink]

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New post 26 May 2019, 17:47
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For how many integers n, such that 2≤n≤80, \(\frac{(n-1)*n*(n+1)}{8}\) is an integer.

A. 9
B. 29
C. 40
D. 49
E. 59
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Re: For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/  [#permalink]

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New post 26 May 2019, 18:09
nick1816 wrote:
For how many integers n, such that 2≤n≤80, \(\frac{(n-1)*n*(n+1)}{8}\) is an integer.

A. 9
B. 29
C. 40
D. 49
E. 59


I started putting random numbers to find a pattern so that I can extrapolate.

I observed that for every (n-1)*n*(n+1) where n is a multiple of 8 the function f(n) is perfectly divisible by 8.
Also, n that is odd, f(n) is also perfectly divisible by 8.

So, there are 10 multiples of 8 between [2,80].
And there are 39 odd numbers between [2,80].
So possible values of n (Integers) 49, IMO D
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For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/  [#permalink]

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New post 26 May 2019, 19:48
nick1816 wrote:
For how many integers n, such that 2≤n≤80, \(\frac{(n-1)*n*(n+1)}{8}\) is an integer.

A. 9
B. 29
C. 40
D. 49
E. 59


(n-1)*n*(n+1) is a product of 3 consecutive integers.

So, it can be of 2 types
1. even*odd*even
2. Odd*even*odd

Case1: even*odd*even
If one even number is a multiple of 2, the next even number will definitely be a multiple of 4.
Eg: 2*3*4, 10*11*12 etc..

So, number if possible values of n are
3, 5, . . . . 79.

To find number of terms of Arithmetic Progression, use Last term
79 = 3 + (n-1)2
n = 39.

Case2: Odd*even*odd
In this case, the even number SHOULD be a multiple of 8 as the other 2 numbers are odd.
So, favourable values are just the number of multiples of 8 from 2≤n≤80 which are 10 values.

So, total values of n = 39 + 10 = 49

IMO D.

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For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/  [#permalink]

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New post 26 May 2019, 22:52
(n-1)n(n+1)/8=Integer , where 2<=n=<80

(n^2-1)n/8=Integer ,True only when
(1) n=odd. e.g. (7^2-1)(7)/8=Integer
(2) (n^2-1)n is a multiple of 8 e.g. (8^2-1)8/8 =Integer

Odd numbers =(80-2)/2 =39
Multiples of 8 = (80-8)/8 +1 =10
Therefore total number of integers is 49
Answer D

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For how many integers n, such that 2≤n≤80, [m][fraction](n-1)*n*(n+1)/   [#permalink] 26 May 2019, 22:52
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