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If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is
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Dillesh4096
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If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is [#permalink] New post  23 Sep 2019, 09:30
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If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is the remainder when x+2 is divided by 21!

A. 0
B. 1
C. 2
D. 19
E. 20

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chetan2u
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If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is [#permalink] New post  23 Sep 2019, 18:31
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Dillesh4096 wrote:
If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is the remainder when x+2 is divided by 21!

A. 0
B. 1
C. 2
D. 19
E. 20

Posted from my mobile device


I am sure the question is meant to be ..
\(x = 1*1! + 2*2! + 3*3! + . . . . . . + 20*20!\)

\(x = 1*1! + 2*2! + 3*3! + . . . . . . + 20*20!\)
= \((2-1)*1! + (3-1)*2! + (4-1)*3! + . . . . . . + (21-1)*20!\)
=\(2*1!-1!+3*2!-2!+4*3!-3!+.....21*20!-20!\)
=\(2!-1!+3!-2!+4!-3!+......20!-19!+21!-20!\)
=\(21!-1!\)

So \(x+2=21!-1+2=21!+1..\)
when you divide 21!+1 by 21!, the remainder is 1..
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Re: If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is [#permalink] New post  23 Sep 2019, 21:15
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the trick here is that 20X20! is almost 21!, let's look at it:
20X20! = 20X20!+20!-20! = 21X20!-20! = 21!-20!
19X19! = 20!-19!
18X18! = 19!-18!
...
3X3! = 4!-3!
2X2! = 3!-2!
1X1! = 2!-1!
summing all the terms up for x we get
x = 2+21!-20!+20!-19!+19!-18!+...+3!-2!+2!-1!
most of the terms cancel out and we are left with
x = 2+21!-1 = 21!+1
now it is easy to see that:
x/21! = (21!+1)/21! = 1+1/21! = (1)
so the remainder is 1 and the correct answer is B
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Re: If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is [#permalink] New post  24 Mar 2022, 20:49
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Re: If x = 1x1! + 2x2! + 3x3! + . . . . . . + 20x20!. What is [#permalink]
24 Mar 2022, 20:49
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