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What is the greatest prime factor of 12^3 – 96?

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Bunuel
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What is the greatest prime factor of 12^3 – 96? [#permalink] New post   02 Nov 2020, 04:30
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A
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35% (medium)

Question Stats:

70% (01:45) correct 30% (01:32) wrong based on 83 sessions
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What is the greatest prime factor of \(12^3 – 96\)?

A. 2
B. 3
C. 7
D. 17
E. 19
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D
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rajatchopra1994
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Re: What is the greatest prime factor of 12^3 – 96? [#permalink] New post   02 Nov 2020, 04:43
Explanation:

12^3 - 9^6
3^3 x 4^2 - 3^12
3^3 x (4^2 - 3^9)

it will be divisible by 3 only.

IMO-B
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Re: What is the greatest prime factor of 12^3 – 96? [#permalink] New post   02 Nov 2020, 04:47
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Bunuel wrote:
What is the greatest prime factor of \(12^3 – 9^6\)?

A. 2
B. 3
C. 7
D. 17
E. 19


greatest prime factor of \(12^3 – 9^6\) = greatest prime factor of \((12^3 – 81^3)\)

\((12^3 – 81^3)\) = (12-81)(12^2+81^2+12*81) = -69*...

it's divisible by 23

So I see something wring with options.
Bunuel
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Bunuel
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Re: What is the greatest prime factor of 12^3 – 96? [#permalink] New post   02 Nov 2020, 04:50
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GMATinsight wrote:
Bunuel wrote:
What is the greatest prime factor of \(12^3 – 9^6\)?

A. 2
B. 3
C. 7
D. 17
E. 19


greatest prime factor of \(12^3 – 9^6\) = greatest prime factor of \((12^3 – 81^3)\)

\((12^3 – 81^3)\) = (12-81)(12^2+81^2+12*81) = -69*...

it's divisible by 23

So I see something wring with options.
Bunuel



The question reads 12^3 – 96, not 12^3 – 9^6. Edited. Thank you.
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Re: What is the greatest prime factor of 12^3 – 96? [#permalink] New post   03 Nov 2020, 05:17
rajatchopra1994 wrote:
Explanation:

12^3 - 9^6
3^3 x 4^2 - 3^12
3^3 x (4^2 - 3^9)

it will be divisible by 3 only.

IMO-B


As Question changes!!

12^3 - 96
1728 - 96
1632
2^5 x 3 x 17

So IMO-D
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Re: What is the greatest prime factor of 12^3 – 96? [#permalink] New post   03 Nov 2020, 08:08
Bunuel wrote:
What is the greatest prime factor of \(12^3 – 96\)?

A. 2
B. 3
C. 7
D. 17
E. 19


Split into 12^3 and 96 into primes.
12=2^2×3 96=2^5×3
Use these primes in the equation 12^3-96

(2^2×3)^3 - (2^5×3) Don't forget the power of 12 [in bold]

2^6×3^3 - 2^5×3

2^5×3 (2×3^2 - 1×1) Take common terms outside.

2^5×3 (18 - 1)

2^5×3 (17)

17 is the greatest prime. Thus option D.

Thank you :)

PS:- While solving I used the basic multiplication method and it consumed a good amount of time. This was comparatively much faster.

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Re: What is the greatest prime factor of 12^3 – 96? [#permalink]
03 Nov 2020, 08:08
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