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the difference between a positive integer, n, and its cube is 4896. Compute n. Please give solution and detailed explanation! Thank you ver much! I tried and got 17, but what i did is to try numbers one by one, so i would really appreciate if anyone can tell me the right and systematic way to tackle this question??

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    $\begingroup$ What if someone asks "Product of three successive integers is $4896$. What is the middle number?"? Hint: Prime factorization of $4896$. $\endgroup$ – Alistair Nov 4 '14 at 2:15
  • $\begingroup$ You can factor 4896 but since (n-1)n(n+1) is close to the cube root of 4896 and started from that. Rounded 16.5+ to 17 and showed that 16*17*18 equals 4896. Although I guessed n = 17 it was an easy and fast guess to fit the three successive factors formula. $\endgroup$ – K7PEH Nov 4 '14 at 2:18
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$\textbf{HINT-}$ you have to solve $n^3-n-4896=0$. $n=17 $ is the only real solution to this cubic. So there are none other.

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  • $\begingroup$ @Annie you can accept in and upvote if it is upto your satisfaction. thanks $\endgroup$ – Bhaskar Vashishth Nov 4 '14 at 19:37
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Like what k7peh has said you can break it down into the product of three consecutive number. There you fact the number. Since 17 is the only prime factor, the rest of the factors are exact to multiply into 16 and 18. You can go from there. It is quite easy because I did it on a division contest like a week ago.

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