What is the remainder when the positive integer n is divided by the positive integer k, where k>1?
(1) n = (k+1)^3
(2) k = 5
GMAT prep says (1) is sufficent to answer the question.
I wonder how????
Am I missing something?
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good analytical approach FULL MARKS 8)
sigh
i hv forgotten all .. i used other approach like any number that is just greater than one than the other number is not divisible by it and the remainder is always one.
then we have condition A
good
sigh
i hv forgotten all .. i used other approach like any number that is just greater than one than the other number is not divisible by it and the remainder is always one.
then we have condition A
good
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To distribute:
(k+1)^3 = (k+1)(k+1)(k+1)
let's take the first two sets of parentheses first:
(k+1)(k+1) = k^2 + 2k + 1
now, multiple in our third (k+1):
(k+1)(k^2 + 2k + 1) =
k^3 + 2k^2 + k + k^2 + 2k + 1) =
k^3 + 3k^2 + 3k + 1 = n
They ask about n/k, so:
(k^3 + 3k^2 + 3k + 1)
k
= k^2 + 3k + 3 + 1/k. In this algebraic representation of division, the numerator of the fraction is the remainder. So the remainder is always 1.
You can also try some real numbers and see if there's a pattern. We know they're both positive integers and k>1.
Try k = 2. Then n = (2+1)^3 = 27. 27/2 = 13 R1
Try k = 3. Then n = (3+1)^3 = 64. 64/3 = 21 R1
So the first thing I notice is that, when k is even, n is odd, and vice versa. The second thing I notice is that I've gotten R1 for both. Interesting.
k = 4. Then n = (4+1)^3 = 125. 125/4 = 31 R1. R1 again.
k = 5. Then n = (5+1)^3 = 216. 216/5 = 43 R1. R1 yet again!
At this point I'm feeling pretty comfortable that there's a pattern. I'm going to get R1 no matter what I try for k.
(k+1)^3 = (k+1)(k+1)(k+1)
let's take the first two sets of parentheses first:
(k+1)(k+1) = k^2 + 2k + 1
now, multiple in our third (k+1):
(k+1)(k^2 + 2k + 1) =
k^3 + 2k^2 + k + k^2 + 2k + 1) =
k^3 + 3k^2 + 3k + 1 = n
They ask about n/k, so:
(k^3 + 3k^2 + 3k + 1)
k
= k^2 + 3k + 3 + 1/k. In this algebraic representation of division, the numerator of the fraction is the remainder. So the remainder is always 1.
You can also try some real numbers and see if there's a pattern. We know they're both positive integers and k>1.
Try k = 2. Then n = (2+1)^3 = 27. 27/2 = 13 R1
Try k = 3. Then n = (3+1)^3 = 64. 64/3 = 21 R1
So the first thing I notice is that, when k is even, n is odd, and vice versa. The second thing I notice is that I've gotten R1 for both. Interesting.
k = 4. Then n = (4+1)^3 = 125. 125/4 = 31 R1. R1 again.
k = 5. Then n = (5+1)^3 = 216. 216/5 = 43 R1. R1 yet again!
At this point I'm feeling pretty comfortable that there's a pattern. I'm going to get R1 no matter what I try for k.
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Stacey Koprince
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Stacey Koprince
GMAT Instructor
Director of Online Community
Manhattan GMAT
Contributor to Beat The GMAT!
Learn more about me