Bigtorial

Number Theory Level 1

Find the last digit of $3665456327618653!$ .

Notation: $!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

3 9 5 0

2 solutions

Akash Patalwanshi
May 14, 2017

If $n ≥ 5$ , where $n$ is natural number then, $n!$ ends with at least one zero

Chris Rather not say
May 7, 2017

Because 3665456327618653>10 the factorial will have a time 10 term and any number times 10 has the last digit of 0 so therefore the last digit of 3665456327618653! is 0 ;)

The number $n$ whose factorial $n!$ contains the units digit as zero does not have its smallest bound on $10$ , i.e., although it is true that if $n > 10$ then its units digit will be zero, the number $10$ is not the smallest value satisfying this inequality.

Q: Can you provide me the lowest bound, i.e, the smallest positive integer $k$ such that $n \ge k$ for which $n!$ has its units digit as zero? Also state the reason why $k$ is the smallest such possible value.

Tapas Mazumdar - 4 years, 1 month ago

Answer:- $k$ is $5$

Reason:- $10$ on prime factorisation give $5×2$ . Therefore the smallest number having $5,2$ both as its factors will be the answer.

Btw why r u asking so easy question. Think about the rightmost non-zero digit of that factorial.

Rahil Sehgal - 4 years, 1 month ago

Good point and so it should be 3665456327618653>5 not 10

Chris Rather not say - 4 years, 1 month ago

The question was intended for the one who posted the solution. As for the rightmost non-zero digit goes, it is $8$ .

Tapas Mazumdar - 4 years, 1 month ago

@Tapas Mazumdar – I have another question for that person. Find the number of attached zeros in that factorial @Chris Rather not say

Rahil Sehgal - 4 years, 1 month ago

@Tapas Mazumdar – Clap Clap Clap

Chris Rather not say - 3 years, 6 months ago

since more than 5, the last digit after 8 is zero........lots of zero...

Ramiel To-ong - 4 years ago

Bigtorial

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