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For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 01:23
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48% (01:51) correct 53% (02:08) wrong based on 39 sessions
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For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is A. 6 B. 24 C. 36 D. 48 E. 96 Competition Mode Question
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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 02:12
(n–1)*n*(n+1)(5n + 2) n=2 > 1*2*3*12 divisible by 6, 24 and 36 n=3 > 2*3*4*17 divisible by 6, 24, but NOT 36
FINAL ANSWER IS (B) 24



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 03:30
Quote: For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is
A. 6 B. 24 C. 36 D. 48 E. 96 The smartest way in such problems is to substitute values and observe the results. Always try with three consecutive values @n=1, n(n^2 – 1)(5n + 2) = 0 @n=2, n(n^2 – 1)(5n + 2) = 2*3*12 = 72 @n=3, n(n^2 – 1)(5n + 2) = 3*8*17 = 24*17 @n=4, n(n^2 – 1)(5n + 2) = 4*15*22 = 24*5*11 @n=5, n(n^2 – 1)(5n + 2) = 5*24*27 The common and highest number (GCD) that divides all the results obtained = 24 Answer: Option B
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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 03:56
Substitute with the smallest positive integer where the answer is not 0 when we use 2 we get the answer 72 and highest divisible integer is 36
(c)



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 04:29
Solution: n (n^2 – 1) (5n + 2) = (n – 1), n,(n + 1), (5n +2) Putting n = 2, 3, 4, 5,…………. n =2, (n – 1)* n*(n + 1)*(5n +2) = 1* 2* 3* 12, divisible by 24
n = 3, (n – 1)* n*(n + 1)*(5n +2) =2* 3* 4 *17, divisible by 24
n = 4, (n – 1)* n*(n + 1)*(5n +2) = 3*4*5*22, divisible by 24
n =5, (n – 1)* n*(n + 1)*(5n +2) = 4*5*6*27, divisible by 24 Answer : B



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 04:34
For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is
A. 6 B. 24 C. 36 D. 48 E. 96
(n1)*n*(n+1) > 3 consecutive numbers > will always be divisible by 6
5n+2 can be 7,12,17 etc
So the equation will always be divisible by 6 as the highest number
Answer  A



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 07:12
Quote: For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is
A. 6 B. 24 C. 36 D. 48 E. 96 n is any positive integer n(n^2 – 1)(5n + 2) n(n+1)(n1)(5n + 2) for n=odd=1: 1(1+1)(11)(5+2)=0 for n=odd=3: 3(4)(2)(17) for n=even=2: 2(3)(1)(12) gcf(for n: 3,2): 3*2*4=24 note: product of three consecutive integers is divisible by at least two evens, and one odd. Ans (B)



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 08:25
n(n^2 – 1)(5n + 2) ; n( n+1)(n1)(5n+2)
for n =1 ; we get 0 n=2; 2*3*1*12 ; 2^3*3^2 n=3 ; 2^3*3*17 common is for all values 2^3*3 ; 24 IMO B ; 24
For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is
A. 6 B. 24 C. 36 D. 48 E. 96



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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 10:18
For every positive integer n, the highest number that n(n^2 – 1)(5n + 2) is always divisible by is A. 6 B. 24 C. 36 D. 48 E. 96 \(n(n^2 – 1)(5n + 2)\) = (n – 1)n(n + 1)(5n + 2) which suggests that it is always divisible by 3 and 2 since (n – 1)n(n + 1) is a multiple of three consecutive numbers. Hence 6 is the highest number that divides \(n(n^2 – 1)(5n + 2)\) always Also, if least values of n are considered, the highest number can be found out. For 1*2*3... , 2*3*4.. , 3*4*5... 6 is the number that divides the \(n(n^2 – 1)(5n + 2)\) always. Answer A.
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Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 21:31
X:n(n^2 – 1)(5n + 2)
starting with n=1....X=0 n=2....X=2(3)(12)=24*3 n=3....X=3(8)(17)=24*17 n=4....X=4(15)(22)= 24*55
the greatest common divisor is 24
OA:B




Re: For every positive integer n, the highest number that n(n^2 – 1)(5n +
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19 Feb 2020, 21:31






