I was born in 1996!

Number Theory Level 3

I was born in year 1996. Find the number of trailing zeroes in $1996!$ .

Clarification:
It is $1996$ factorial. That is $1996 \times 1995 \times 1994 \times \ldots \times 2\times 1$ .

The answer is 496.

5 solutions

Jaydee Lucero
Aug 5, 2014

Legendre's theorem. $\left[\frac{1996}{5}\right]+\left[\frac{1996}{25}\right]+\left[\frac{1996}{125}\right]+\left[\frac{1996}{625}\right]+\text{ ... }=496$ Here, $[x]$ represents the greatest integer less than or equal to the real number $x$ .

hi. i wrote 10 in place of 5 so that i got 219 10,s in the factorial. then i thought that 219 zeroes in the end of the factorial's value. why isn't my method correct? please help me out here.

Lipsa Kar - 5 years, 9 months ago

10 will "miss" half of the 5s, namely all of the 5s multiplied by odds (like 15, 35, 45). Let me take it back a step, the zeros are 10s, but the prime factorization of 10 is 2*5. You quickly find that this problem, when factored by primes, has an over abundance of 2s, and that the number of 5s dictates the logic. The first sweep of dividing 1996 by 5 yields 399 5s. But then one needs to take into account the numbers in 1996! that have 5^2 in their factorizations, and then 5^3, etc.
To answer your question, 10 misses a bunch of 5s.

Ken Hodson - 5 years, 6 months ago

Dollesin Joseph
Aug 8, 2014

1996/5 = 399

399/5=79

79/5=15

15/3=3

399+79+15+3 = 496

Why 5 is divided

Subrat Panigrahi - 6 years, 10 months ago

Because you need to count how many times 1996! is multiplied to 10 or in other words count the number of (5x2) in 1996!. Since 2 is used more than 5 in 1996!, just count 5. :)

Gil Deon Basa - 6 years, 9 months ago

For this ans, I think that the last step should've been 15/5 = 3 and not 15/3 = 3.

Ritu Roy - 6 years, 9 months ago

why 4 terms only?

jayanth raavi - 6 years, 9 months ago

In this process, you are dividing each time by 5 and finding the closest integer value without going over. You could theoretically keep going, but then your fifth term would be 3/5=0 and all subsequent terms would be 0/5=0, so after the fourth term you would be adding 0 to your total infinitely many times. Adding 0 would have no effect on your answer.

Kunal Kantaria - 5 years, 11 months ago

Bill Bell
Jul 19, 2014

I would never manage to get a correct answer to this by hand. In Python:

>>> N=1996

>>> zeroes=0

>>> for i in range(1, int(log(N,5))+1):

... zeroes+=int(1. N/5 *i)

...

>>> zeroes

496

Plz xpln more.Looks lyk a good alternative soln.

Chandrachur Banerjee - 6 years, 9 months ago

Rhiane Sanchez
Jul 11, 2014

5^1 = 1996 ÷ 5 = 399.2

5^2 = 1996 ÷ 25 = 79.84

5^3 = 1996 ÷ 125 = 15.968

5^4 = 1996 ÷ 625 = 3.1936

5^5 = 1996 ÷ 3125 = 0.63872

399 + 79 + 15 + 3 = 496

I got it too!

Daryll RomuaLdez - 5 years, 8 months ago

Astro Enthusiast
Jul 10, 2014

Divide the number by 5. Divide it by 25. Divide it by 125. Divide it by 625.

As long as the answer n < 5, you do not stop dividing the number by 5^x. x is the exponent.

Just get the whole numbers from the quotients and never mind the fractions.

Why is this becoming over-rated? HAHAHA.

Astro Enthusiast - 6 years, 9 months ago

Yes.....so highly overrated!

Krishna Ar - 6 years, 9 months ago

Wait, Is your name really Precious Prestosa?

Krishna Ar - 6 years, 11 months ago

Yes, HAHAHA.

Astro Enthusiast - 6 years, 11 months ago

Wow! That's (Precious) a nice first name :P

Krishna Ar - 6 years, 11 months ago

@Krishna Ar – HAHA. Thank you ;)

Astro Enthusiast - 6 years, 11 months ago

@Astro Enthusiast – This is a nice solution Precious ;)

Abi Krishnan - 6 years, 10 months ago

@Astro Enthusiast – How did you change your name?

Satvik Golechha - 6 years, 8 months ago

I was born in 1996!

The answer is 496.

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