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02 Dec 2018, 23:52
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
71% (02:23) correct
29%
(02:27)
wrong
based on 426
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If \(n\) is a positive integer, which of the following could be the value of \((n+1)^3 - n^3\)?
A. 629
B. 630
C. 631
D. 632
E. 633
A. 629
B. 630
C. 631
D. 632
E. 633
05 Dec 2018, 02:08
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Recall that \(x^3 – y^3 = (x-y)(x^2+xy+y^2)\).’
Now,
\((n+1)^3 - n^3\)
\(= ( n + 1 – n ) ( (n+1)^2 + (n+1)n + n^2 )\)
\(= (n+1)^2 + (n+1)n + n^2\)
\(= n^2 + 2n + 1 + n^2 + n + n^2\)
\(= 3n^2 + 3n + 1\)
\(= 3n(n+1) + 1\)
Thus \((n+1)^3 - n^3\) has remainder \(1\) when it is divided by \(3\).
\(631\) is the only answer choice with remainder \(1\) when it is divided by \(3\).
Therefore, the answer is C.
Answer: C
Recall that \(x^3 – y^3 = (x-y)(x^2+xy+y^2)\).’
Now,
\((n+1)^3 - n^3\)
\(= ( n + 1 – n ) ( (n+1)^2 + (n+1)n + n^2 )\)
\(= (n+1)^2 + (n+1)n + n^2\)
\(= n^2 + 2n + 1 + n^2 + n + n^2\)
\(= 3n^2 + 3n + 1\)
\(= 3n(n+1) + 1\)
Thus \((n+1)^3 - n^3\) has remainder \(1\) when it is divided by \(3\).
\(631\) is the only answer choice with remainder \(1\) when it is divided by \(3\).
Therefore, the answer is C.
Answer: C
General Discussion
03 Dec 2018, 01:23
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\((n+1)^3-n^3\) = \(n^3+1+3n(n+1)-n^3\) = 1+3n(n+1)
Consider n as 15 then 1+3n(n+1) = 721
now, n should be less than 15. Take 13, 1+3n(n+1) = 547
Only possible option is 14, then 1+3n(n+1) = 631.
C is the answer.
Consider n as 15 then 1+3n(n+1) = 721
now, n should be less than 15. Take 13, 1+3n(n+1) = 547
Only possible option is 14, then 1+3n(n+1) = 631.
C is the answer.
