If n is a positive integer, which of the following could be

[Max@Math Revolution]

[Math Revolution GMAT math practice question]

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

[fskilnik@GMATH]

[Math Revolution GMAT math practice question]

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

$$n \ge 1\,\,{\mathop{\rm int}} \,\,\left( * \right)$$
$$?\,\,:\,\,\,{\left( {n + 1} \right)^3} - {n^3}\,\,\underline {{\rm{could}}} \,\,{\rm{be}}$$

$${\left( {n + 1} \right)^3} - {n^3}\,\,\, = \,\,\, \ldots \,\,\, = \,\,\,3{n^2} + 3n + 1\,\,\, = \,\,\,3\left( {{n^2} + n} \right) + 1\,\,\, = \,\,\,3Q + 1\,\,\,,\,\,\,\,Q\mathop \ge \limits^{\left( * \right)} 2\,\,\,{\mathop{\rm int}} $$

$$?\,\,\,\,:\,\,\,{\rm{remainder}}\,\,{\rm{1}}\,\,{\rm{when}}\,\,{\rm{divided}}\,\,{\rm{by}}\,\,{\rm{3}}\,\,\,\,\, \Rightarrow \,\,\,{\rm{only}}\,\,\left( {\rm{C}} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Jay@ManhattanReview]

[Math Revolution GMAT math practice question]

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

Among the five options, we have to find out the possible value of (n+1)^3 - n^3.

So, we have (n+1)^3 - n^3

= n^3 + 3n(n + 1) + 1 - n^3

= [3n(n + 1)] + 1

Since is n a positive integer and n and (n + 1) are consecutive integers, one of n and (n + 1) is even. Thus, we can conclude that 3n(n + 1) is even and a multiple of 3*2 = 6. Thus, Between the two even numbers 630 and 632, only 630 is divisible by 6. So, 630 + 1 = 631 must the correct answer.

The correct answer: C

Hope this helps!

-Jay
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[Max@Math Revolution]

=>

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 + n2
= n^2 + 2n + 1 + n^2 + n + n2
= 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