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Calculate the units digit of the following expression:

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Calculate the units digit of the following expression:  [#permalink]

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New post Updated on: 13 Aug 2018, 02:35
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A
B
C
D
E

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  45% (medium)

Question Stats:

65% (01:26) correct 35% (01:42) wrong based on 169 sessions

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e-GMAT Question:



Calculate the units digit of the following expression:

\(1!^1 + 2!^2 + 3!^3 +4!^4 +5!^5 ….. + 10!^{10}\)

    A) 1
    B) 3
    C) 5
    D) 7
    E) 9

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Question 1 of The e-GMAT Number Properties Marathon




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Question 2 of the Marathon


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Originally posted by EgmatQuantExpert on 27 Feb 2018, 09:52.
Last edited by EgmatQuantExpert on 13 Aug 2018, 02:35, edited 3 times in total.
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Re: Calculate the units digit of the following expression:  [#permalink]

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New post 27 Feb 2018, 10:03
D) 7

In the expression fifth onward terms will have unit digits 0,
Fourth term's unit digit will be 6,
Third term's unit digit will be 6,
Second term's unit digit will be 4,
First term's unit digit will be 1,
Hence the unit digit of the expression will be given by
0+6+6+4+2+1=17 which has 7 as units digit.


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Re: Calculate the units digit of the following expression:  [#permalink]

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New post 27 Feb 2018, 22:57

Solution



    • The units digit of the expression will depend on the units digits of the factorials and on the powers of each term
    • To simplify things, let us temporarily focus on the units digit of the factorials and then we will move on to their exponents.
    • While simplifying the factorials of the numbers we will observe the following:
      o \(1! = 1\)
      o \(2! = 2\)
      o \(3! = 6\)
      o \(4! = 24\): Units digit =\(4\)
      o \(5! = 120\): Units digit =\(0\)
      o We do not need to calculate any further since after \(4!\), the value of the next terms will end with zeroes.
         Example: \(6! = 5!*6 = 120*6 = 720\) : Units digit: \(0\)
         This can be logically deduced because we have a \(5\) and \(2\) in each factorial starting from \(5!\), which gives a zero at the end every time.
    • Since zero added to any number yield the same number, we can ignore it and in turn ignore all terms from \(5! ^5\) to \(10! ^{10}\).
    • This method helped us eliminate all the terms after \(4!\) and we can re-write the whole expression as shown below:
    \(1! ^1 + 2!^ 2 + 3! ^3 + 4! ^4\).
    Now we can calculate the units digit of each of the factorials and find out the units digit of our main expression.

    • Units digit of \(1! ^1\):
      o We know that the cyclicity of \(1\) is \(1\).
      o Thus, units digit is \(1\)
    • Units digit of \(2! ^2\):
      o We know \(2^2 = 4\), thus the units digit is \(4\)
    • Units digit of \(3! ^3\) or \(6^3\):
      o Cyclicity of \(6\) is \(1\)
      o Thus, units digit of \(6^3\) = \(6\)
    • Units digit of \(4!^ 4\) or \(24^4\):
      o Cyclicity of \(4\) is \(2\).
      o The given number is of the form \(4^{2k}\), where “k” is any natural number. Hence the units digit =\(6\)
    The units digit of the expression will be the sum of all the individual unit digits calculated above.
    • Units digit of \(1!^1 + 2!^2 + 3!^3 + 4!^4\) = \(1 + 4 + 6 + 6 = 17\), which implies the units digit is \(7\).
Since, the units digit of the given expression is 7, and the correct answer is Option D.
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Re: Calculate the units digit of the following expression:   [#permalink] 27 Feb 2018, 22:57
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