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b, c, and d are consecutive even integers such that 2 < b < c < d. Wha

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b, c, and d are consecutive even integers such that 2 < b < c < d. Wha  [#permalink]

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New post 28 Jan 2019, 00:28
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b, c, and d are consecutive even integers such that 2 < b < c < d. What is the largest positive integer that must be a divisor of bcd?

A. 12
B. 16
C. 24
D. 48
E. 96

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Re: b, c, and d are consecutive even integers such that 2 < b < c < d. Wha  [#permalink]

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New post 28 Jan 2019, 01:54
Bunuel wrote:
b, c, and d are consecutive even integers such that 2 < b < c < d. What is the largest positive integer that must be a divisor of bcd?

A. 12
B. 16
C. 24
D. 48
E. 96



Note: b, c, d are consecutive even integer****

let's put some values for b,c,d.

b<c<d = 4<6<8.

bcd = 72.

72 is not divisible by 96 , 48 and 16.

we are left with 24 and 12.

b=6
c=8
d=10

bcd = 6*8*10 = 480.

divisible by both 12 and 24.

So, 24 is our answer.

Largest value that divide into any combination of bcd.

C is the correct answer.
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Re: b, c, and d are consecutive even integers such that 2 < b < c < d. Wha  [#permalink]

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New post 28 Jan 2019, 01:57
1
Bunuel wrote:
b, c, and d are consecutive even integers such that 2 < b < c < d. What is the largest positive integer that must be a divisor of bcd?

A. 12
B. 16
C. 24
D. 48
E. 96



b, c and d are consecutive even integers such that 2<b<c<d.
Let's assume b = 2n, then c = 2n+2 , d = 2n+4 and n>1 since 2<b<c<d.

Then the expression bcd = 2n(2n+2)(2n+4)= 8n(n+1)(n+2). Now n(n+1)(n+2) is the product of 3 consecutive integers which is always divisible by 6.
Therefore 8n(n+1)(n+2) is always divisible by 6*8 or 48
IMO D
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Re: b, c, and d are consecutive even integers such that 2 < b < c < d. Wha  [#permalink]

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New post 28 Jan 2019, 20:57
three consecutive integers are: 2k-2, 2k, 2k+2
if you take 2 out from each of the numbers:
8(k-1)(k)(k+1)

Also according to the question:
2k-2>2
2k>4
k>2

Now
Lets take any values:
234
345
456
567(only divisible by 2*)

Every three consecutive numbers is divisible by at least 2* and every 3 consecutive integers is divisible by 3.

Hence the maximum divisor for every variable bcd: 2*3*8=48

Hence Ans D
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Re: b, c, and d are consecutive even integers such that 2 < b < c < d. Wha   [#permalink] 28 Jan 2019, 20:57
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b, c, and d are consecutive even integers such that 2 < b < c < d. Wha

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