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# n=5*(10!)

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n=5*(10!)

no of zeros does n end with?

Guest Apr 1, 2017
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#1
+1

If you want to know the number of times the exprssion 'x' is divisible by 'a' you need to find a's prime factors and count the times x is divisible by each one of them.

You probably have no idea what im talking about so ill give an example:

Lets see how many times 9! Is divisible by 3.

3's only prime factor is 3. So we need to count the times 9! Is divisible by 3.

There are 3 numbers that are divisible by 3 and arent bigger than 9: 3, 6, 9. 3 is divisible by 3 once, 6 is divisible by 3 once and 9 is divisible by 3 twice. 1+1+2=4. So 9! Is divisible by 3 4 times.

But 3 is a prime and he has only one prime factor. Lets see what happens when we have more that one factor shall we?

Lets take 6. 6's prime factors are 3 (once) and 2 (once).

The new question is how many times is 12! Divisible by 6?

The first thing we have to do is count the times 12! Is divisible by 2. Im too lazy so i wont write the whole process again. The answer is 10. For 3, the answer is 5.

BUT WE GOT 2 DIFFERENT ANSWERS W*F WHAT SHOULD WE DO??!???!11!?!

Well, we need to pick the smaller number, which is 5. Why? Lets think about it in a different way. We got 3^5 and 2^10. We need to count the times 3^5*2^10 is divisible by 6.

3^5*2^10=(2*3)^5*2^5. We dont care about the 2^5 that got out because it is not divisible by 3, therefore not divisible by 6

Guest Apr 1, 2017
#2
+25984
+2

A more pictorial approach!

This wouldn't be so good for much bigger numbers though!

.

Alan  Apr 1, 2017
edited by Alan  Apr 1, 2017
edited by Alan  Apr 1, 2017
#3
0

Isn't the question about the number of trailing zeros in:

5 x 10!? =18,144,000?

Guest Apr 1, 2017
#4
+310
+1

The number of zeroes is the number of times it is divisible by 10. think about it. 5*10=50. we Multiplied by 10, and got a new zero. multiplying it again will give us another zero.

Ehrlich  Apr 1, 2017
#5
0

Here is a "Rule of Thumb" when trying to find the number of trailing zero of a large factorial:

Example: How many trailing zeros are there in 5,000! ???.

You simply divide the number in question by "powers of 5", as follows:

5,000/5 =1,000

5,000/25=200

5,000/125 =40

5,000/625 =8

5,000/3,125 =1, when there a fraction involved, just ignore the fraction and take the integer part.

Total =1,000+200+40+8+1 =1,249 trailing zeros in 5,000!.

Guest Apr 1, 2017
#6
+18369
0

n=5*(10!)

no of zeros does n end with?

Legendre's Theorem - The Prime Factorization of Factorials

In mathematics,

Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial

$${\displaystyle n!}$$

1. The exponent of 2x

We calculate 10 in base 2:

$$\begin{array}{rcll} 10_{10} &=& 1010_2 \\ \end{array}$$

The sum of the digits in base 2 is $$\begin{array}{rcll} 1+0+1+0 = 2 \end{array}$$

The exponent is: $$\begin{array}{|rcll|} \hline x &=&\frac{10-(\text{sum of the standard base-}\mathbf{2}\text{ digits of 10})}{\mathbf{2}-1} \\ x &=&\frac{10-2}{2} \\ x &=&4\\ \hline \end{array}$$

2. The exponent of 5x

We calculate 10 in base 5:

$$\begin{array}{rcll} 10_{10} &=& 20_5 \\ \end{array}$$

The sum of the digits in base 5 is $$\begin{array}{rcll} 2+0 = 2 \end{array}$$

The exponent is: $$\begin{array}{|rcll|} \hline x &=&\frac{10-(\text{sum of the standard base-}\mathbf{5}\text{ digits of 10})}{\mathbf{5}-1} \\ x &=&\frac{10-2}{4} \\ x &=&2\\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 10! &=& 2^4\cdot 5^2\ldots \\ n=5\cdot 10! &=& 2^4\cdot 5^3\ldots \\ \hline \end{array}$$

Number of zeros does n end with ?:

$$\begin{array}{|rcll|} \hline && 2^4\cdot 5^3 \\ &=& 2^3\cdot5^3\cdot 2 \\ &=& (2\cdot 5)^3 \cdot 2 \\ &=& 2\cdot 10^\mathbf{\color{red}3} \\ \hline \end{array}$$

n ends with 3 zeros.

heureka  Apr 3, 2017

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