60!

Every time you have 5 x 2 you get a zero at the end of a number. This is also the ONLY time you will get a zero at the end. As there are many more 2s than there are 5s, the number of zeros will be equal to the number of prime factors of 5 that 60! has. Here are the counts:


5 - 1
10 - 1
15 - 1
20 - 1
25 - 2
30 - 1
35 - 1
40 - 1
45 - 1
50 - 2
55 - 1
60 - 1

Total them up to 14.

By mistake I counted as 12

60,50,40,30,20,10 = 6
55,45,35,25,15,5 = 6

But then realized than 25 has 2 - 5's & 50 has 2 - 5's so both has 2 extra 5's . counting total to 14.

Good Question.

If 60! is written out as an integer, with how many consecutive 0’s will that integer end?
6
12
14
42
56

60! = 60*59*58*....*3*2*1.

Since 10=2*5, EVERY COMBINATION OF 2*5 contained within the prime-factorization of 60! will yield a 0 at the end of the integer representation of 60!.
The prime-factorization of 60! is composed of FAR MORE 2'S than 5's.
Thus, the number of 0's depends on the NUMBER OF 5's contained within 60!.

To count the number of 5's, simply divide increasing POWERS OF 5 into 60.

Every multiple of 5 within 60! provides at least one 5:
60/5 = 12 --> twelves 5's.
Every multiple of 5² provides a SECOND 5:
60/5² = 2 --> two more 5's.
Thus, the total number of 5's contained within 60! = 12+2 = 14.

The correct answer is C.

Another example:

If 200! is written out as an integer, with how many consecutive 0’s will that integer end?

Every multiple of 5 within 200! provides at least one 5:
200/5 = 40 --> forty 5's.
Every multiple of 5² provides a SECOND 5:
200/5² = 8 --> eight more 5's.
Every multiple of 5³ provides a THIRD 5:
200/5³ = 1 --> one more 5.
Thus, the total number of 5's contained within 200! = 40+8+1 = 49.

Every multiple of 5 within 200! provides at least one 5:
200/5 = 40 --> forty 5's.
Every multiple of 5² provides a SECOND 5:
200/5² = 8 --> eight more 5's.
Every multiple of 5³ provides a THIRD 5:
200/5³ = 1 --> one more 5.
Thus, the total number of 5's contained within 200! = 40+8+1 = 49.

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Hi Mitch ,

Can you please explain the above part.

Also 200/5³ is not perfectly divisible so how come ?

Please explain.

Many thanks in advance.

SJ

Hi ,

Thanks for your reply.

One more thing can we also do 200/5^4 ?

Please advise .

Thanks,

SJ

jain2016 wrote:One more thing can we also do 200/5^4 ?

What does 25 x 25 equal?

Now, answer your own question.

What does 25 x 25 equal?

Now, answer your own question.

[/quote]

Hi Marty ,

25X25 = 625 and I got the my answer.

Thanks for the direction.

SJ