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Re: Calculate the units digit of the following expression:
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28 Feb 2018, 00:46

Solution:

• The units digit of the expression will depend on the units digits of the factorials • 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 \(5!\) to \(10!\). • This method helped us eliminate all the terms after \(4!\) and we can re-write the whole expression as shown below: \(1! + 2! + 3! + 4!\) • To find out the units digit of this expression, let us calculate the value of each term.

o \(1! = 1\) o \(2! = 2\) o \(3! = 6\) o \(4! = 24\)

• Hence, \(1! + 2! + 3! + 4! = 33\). • Units digit of \(33\) =\(3\).

Since the units digit of the given expression is \(3\), and the correct answer is Option B.
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