Number System Tutorial Part III: Factors, Multiples, Unit Digit and Last Two Digits of Exponents

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Factorial (!)

Factorial is a well defined operator, which is defined as the "the product of a certain number of natural numbers starting from 1" The operator factorial is denoted by the symbol '!' (Exclamatory sign).
Examples: `4! = 1 xx 2 xx 3 xx 4 = 24`
`5! = 1 xx 2 xx 3 xx 4 xx 5 = 120`
`n! = 1 xx 2 xx 3 xx "...." xx (n - 2) xx (n - 1) xx n`

Pre defined values
`0! = 1`
`1! = 1`
`n! = n xx (n - 1)! = n xx (n - 1) xx (n - 2)!` , and so on.

Concept I: Largest power of a prime in N!

Let's look at these kind of problems with help of couple of examples.

Find the largest power of 2 in 5!?
`5! = 1 xx 2 xx 3 xx 4 xx 5 = = 2^3 xx 3 xx 5`
`:.`the largest power of 2 in 5! is 3.

If the question ask about a largest factorial value, then the above method is not easy to apply. So we have to think about an alternate method.

Long division method:

Find the largest power of 3 in 100!?
Do a long division of the number using the factor specified.

Largest power of 3 in 100! = sum of all quotient in all steps
`-> 33 + 11 + 3 + 1 = 48`.
If `(76!)/(5^n)` is an integer, find the maximum possible value for n.?
If `(76!)/(5^n)` is an integer, then `n` is the largest exponent of 5 in 76!

The largest possible value for `n = 15 + 3 = 18`

Concept II: Largest power of a composite number in N!

Find the largest power of 6 in 15!.?
`6 = 3 xx 2`; Largest prime factor of 6 is 3.
For finding the largest power of 6 in 15!, it is enough to find the largest power of 3 in 15!

Largest power of `n = 5 + 1 = 6`
So the largest power of 6 in 15! is 6

Concept 3: Number of zeros at the end of N!

In the product of first 'n' natural numbers, a 'zero' will get generated as per the combination of 2 and 5. Therefore finding the number of zeros at the end of N!, find the number of 5's in N!.

How many zeros at the end of 25! ?

i.e. There are 5 + 1 = 6 zeros at the end of 25!

How to find unit digit of an exponential expression

The unit digit of the powers of natural numbers follows some interesting patterns. An understanding of the occurrence of unit digit will help you to score some points in your exams. This awareness is not only for using the typical questions from unit digit but also can use as the verification tool for basic arithmetic operations.

 

Unit digit of the different exponents of all numerals follows a certain cyclic order. It is possible to tabulate the cyclic pattern of unit digits in the following manner.

`"Base"^n`Unit DigitGeneral Form of Exponent
`0^n` 0 Any natural number
`1^n` 1 Any natural number
`2^1` 2 4k + 1
`2^2` 4 4k + 2
`2^3` 8 4k + 3
`2^4` 6 4k + 0
`3^1` 3 4k + 1
`3^2` 9 4k + 2
`3^3` 7 4k + 3
`3^4` 1 4k + 0
`4^1` 4 Odd
`4^2` 6 Even
`5^n` 5 Any natural number
`6^n` 6 Any natural number
`7^1` 7 4k + 1
`7^2` 9 4k + 2
`7^3` 3 4k + 3
`7^4` 1 4k + 0
`8^1` 8 4k + 1
`8^2` 4 4k + 2
`8^3` 2 4k + 3
`8^4` 6 4k + 0
`9^1` 9 Odd
`9^2` 1 Even

Let's look at how to use this table with the help of few examples.

Find the unit digit of `1020^34` ?
1020 ends in 0. From the table, there is only one unit digit for the pattern of 0. i.e. Any exponent of a number which ends in 0 has the unit digit '0'.
Therefore the unit digit of `1020^34` is 0.
Find the unit digit of `1542^345` ?
Exponent = 345
`"Rem"[345/4] = 1`, i.e 345 is of the form `4k + 1`
From the pattern of 2 in the table, if the exponent in the form `4k + 1` then unit digit is 2.
Find the unit digit of `3689^3475` ?
From the table, `9^"odd"` will end in 9 itself.
Therefore the unit digit of `3689^3475` is 9.
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