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As we can see we have the product of 4 consecutive integers: (k – 1), k, (k+1), and (k + 2). Thus, the product will always be divisible by 4! = 24, which means that it will be divisible by 6 too.
Answer: A.
Of course we can simply plug-in numbers: since PS question cannot have more than 1 correct answer, then for any k the answer must be the same. For example, if k = 1, then the product becomes 0. 0 divided by 6 gives the remainder of 0 (0 is a multiple of every integer, except 0 itself).
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Re: If k is a positive integer, what is the remainder when (k + 2)(k^3 – k [#permalink]
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24 Jun 2017, 16:15
3
\((k + 2) (k^3 – k)\)
\((k + 2) (k (K^2 - 1))\)
\((k + 2) k (k + 1) (k - 1)\)
\((k - 1) k (k + 1) (k + 2)\)
As we can see that above numbers are 4 consecutive integers, and as we know \(4\) consecutive integers are multiples of \(4! = 4 * 3 * 2 * 1 = 24\)
Hence, as \(24\)is divisible by \(6\), equation will be commpletely divisible by \(6\) and hence reminder will be ZERO.
Hence, Answer is A
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Just for "fun," let's test another k-value Try k = 2 We get: (k + 2)(k³ – k) = (2 + 2)(2³ – 2) = (4)(6) = 24 When we divide 24 by 6 we get 4 with remainder 0 Answer:
Reordering the factors in the expression, we have:
(k - 1)(k)(k + 1)(k + 2), which is a product of 4 consecutive integers. Since the product of n consecutive integers is always divisible by n!, the product of 4 consecutive integers is always divisible by 4! = 24 and hence by 6. Thus, the remainder when (k + 2)(k^3 – k) is divided by 6 is 0.
Answer: A
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Re: If k is a positive integer, what is the remainder when (k + 2)(k^3 – k [#permalink]
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09 Jan 2018, 18:28
Hi All,
This question it perfect for TESTing VALUES (the approach that Abhishek009 used). The question is also based on a subtle Number Property rule about consecutive integers.
(K+2)(K^3 - K) can be rewritten as.... (K+2)(K)(K^2 - 1) = (K+2)(K)(K+1)(K-1)
When multiplying numbers, the 'order' doesn't matter, so you can re-order those 4 'pieces' as... (K-1)(K)(K+1)(K+2)
We're told that K is a positive integer, so what you have here is the product of 4 consecutive integers (they'll either all be positive or it's be 0-1-2-3) and EVERY 4 consecutive integers will include at least two multiples of 2 and a multiple of 3. By extension, this product will be some multiple of (2)(3) = 6, so when you divide the product by 6 you'll get a remainder of 0.
As we can see we have the product of 4 consecutive integers: (k – 1), k, (k+1), and (k + 2). Thus, the product will always be divisible by 4! = 24, which means that it will be divisible by 6 too.
Answer: A.
Of course we can simply plug-in numbers: since PS question cannot have more than 1 correct answer, then for any k the answer must be the same. For example, if k = 1, then the product becomes 0. 0 divided by 6 gives the remainder of 0 (0 is a multiple of every integer, except 0 itself).
ok so you say we hav a product of 4 consecutive integers (k – 1), k, (k+1), and (k + 2) but that one K is not in the form of (k-1) it is just single K :) why ?
for example i have this (k – 1), k, (k+1), and (k + 2) and will take 4 and plug in
so i get (4 – 1), 4, (4+1), and (4 + 2) --- > 3, 4, 5, 6
but if I take 0 for K i get ( 0 – 1), 0, (0+1), and (0+ 2) ---> i get 0 and then remain is 6 ?
As we can see we have the product of 4 consecutive integers: (k – 1), k, (k+1), and (k + 2). Thus, the product will always be divisible by 4! = 24, which means that it will be divisible by 6 too.
Answer: A.
Of course we can simply plug-in numbers: since PS question cannot have more than 1 correct answer, then for any k the answer must be the same. For example, if k = 1, then the product becomes 0. 0 divided by 6 gives the remainder of 0 (0 is a multiple of every integer, except 0 itself).
ok so you say we hav a product of 4 consecutive integers (k – 1), k, (k+1), and (k + 2) but that one K is not in the form of (k-1) it is just single K why ?
for example i have this (k – 1), k, (k+1), and (k + 2) and will take 4 and plug in
so i get (4 – 1), 4, (4+1), and (4 + 2) --- > 3, 4, 5, 6
but if I take 0 for K i get ( 0 – 1), 0, (0+1), and (0+ 2) ---> i get 0 and then remain is 6 ?
thank you
0 is divisible by every integer (except 0 itself): 0/integer = integer. So, 0 divided by 6 gives the remainder of 0.
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