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# What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11

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Manager
Joined: 19 Aug 2009
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What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11  [#permalink]

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04 Nov 2009, 00:07
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35
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Difficulty:

45% (medium)

Question Stats:

72% (02:18) correct 28% (02:12) wrong based on 267 sessions

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What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(1152!)^3 is divided by 1152?

1. 125
2. 225
3. 325
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Joined: 02 Sep 2009
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31 Jan 2010, 06:17
19
6
jeeteshsingh wrote:
Is there a specific approach to tackle this?

No specific approach.

We have the sum of many numbers: $$(1!)^3+ (2!)^3 + (3!)^3 +...+(1152!)^3$$ and want to determine the remainder when this sum is divided by 1152.

First we should do the prime factorization of 1152: $$1152=2^7*3^2$$.

Consider the third and fourth terms:
$$(3!)^3=2^3*3^3$$ not divisible by 1152;
$$(4!)^3=2^9*3^3=2^2*3*(2^7*3^2)=12*1152$$ divisible by 1152, and all the other terms after will be divisible by 1152.

We'll get $$\{(1!)^3+ (2!)^3 + (3!)^3\} +\{(4!)^3+...+(1152!)^3\}=225+1152k$$ and this sum divided by 1152 will result remainder of 225.
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04 Nov 2009, 03:19
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The ans has to be 225. all the terms in the sequence after (3!)^3 are divisible by 1152 and hence remainder is 0. Upto (3!)^3, sum of all numbers, i.e. 1+8+216 = 225 which is the remainder!!
##### General Discussion
Manager
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31 Jan 2010, 05:01
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Is there a specific approach to tackle this?
Manager
Joined: 10 Feb 2010
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12 Feb 2010, 17:45
Nice Explanation!
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Status: I will not stop until i realise my goal which is my dream too
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14 Jul 2011, 02:13
Bunuel....+1 to you...
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16 Jul 2011, 02:13
good question. Thanks Bunuel for sharing the approach for these problem!
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Joined: 23 May 2012
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Re: What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11  [#permalink]

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19 Oct 2012, 03:21
Did .. the same thing as Bunuel..

But took 8 minutes..

In actual exam.. I would have guessed and moved on

What is the source of this problem?
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18 Jul 2013, 05:20
Bunuel wrote:
jeeteshsingh wrote:
Is there a specific approach to tackle this?

No specific approach.

We have the sum of many numbers: $$(1!)^3+ (2!)^3 + (3!)^3 +...+(1152!)^3$$ and want to determine the remainder when this sum is divided by 1152.

First we should do the prime factorization of 1152: $$1152=2^7*3^2$$.

Consider the third and fourth terms:
$$(3!)^3=2^3*3^3$$ not divisible by 1152;
$$(4!)^3=2^9*3^3=2^2*3*(2^7*3^2)=12*1152$$ divisible by 1152, and all the other terms after will be divisible by 1152.

We'll get $$\{(1!)^3+ (2!)^3 + (3!)^3\} +\{(4!)^3+...+(1152!)^3\}=225+1152k$$ and this sum divided by 1152 will result remainder of 225.

Hi Bunnel,
To get the remainder, we dont have to reduce the fraction right?
That is we cant do - 225/1152 = 25/ 128 and get remainder 25?
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21 Jul 2013, 01:24
cumulonimbus wrote:
Bunuel wrote:
jeeteshsingh wrote:
Is there a specific approach to tackle this?

No specific approach.

We have the sum of many numbers: $$(1!)^3+ (2!)^3 + (3!)^3 +...+(1152!)^3$$ and want to determine the remainder when this sum is divided by 1152.

First we should do the prime factorization of 1152: $$1152=2^7*3^2$$.

Consider the third and fourth terms:
$$(3!)^3=2^3*3^3$$ not divisible by 1152;
$$(4!)^3=2^9*3^3=2^2*3*(2^7*3^2)=12*1152$$ divisible by 1152, and all the other terms after will be divisible by 1152.

We'll get $$\{(1!)^3+ (2!)^3 + (3!)^3\} +\{(4!)^3+...+(1152!)^3\}=225+1152k$$ and this sum divided by 1152 will result remainder of 225.

Hi Bunnel,
To get the remainder, we dont have to reduce the fraction right?
That is we cant do - 225/1152 = 25/ 128 and get remainder 25?

Yes, 225 divided by 1152 yields the remainder of 225. The same way as 2 divided by 4 yields the remainder of 2, not 1 (1:2).
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Re: What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11  [#permalink]

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12 Aug 2015, 11:22
starting from 4!^3=24^3 all numbers are divisible by 1152, i.e. remainder equal to 0

Only 1!^3, 2!^3 and 3!^3 are not divisible by 1152 and have remainder equal to 1,8 and 216, respectively.

If sum numbers we can sum their remanders to find total remainder, which is 216+8+1+0=225

B
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Re: What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11  [#permalink]

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01 Dec 2017, 05:06
factorize 1152 into 2^7*3^2

see carefully all are cubic factorial

we need to look beyonf (4!)^3 as from here onwards remainder is zero

so consider cubic factorial of 1 , 2 and 3 and sum will give 225. and this number on division by 1152 gives 225.

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Re: What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11  [#permalink]

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21 Mar 2020, 05:13
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Re: What is the remainder when (1!)^3+ (2!)^3 + (3!)^3 +.....(11   [#permalink] 21 Mar 2020, 05:13