Question

First ten even natural numbers are multiplied with the first five multiples of five.Find out the number of zeroes at the end of the product

Answer 1

There will be 8 zeroes at the end of product

1) The first ten even natural numbers are: 2,4,6,8,10,12,14,16,18,20

2) First five multiples of 5 are: 5,10,15,20,25

3) Multiplying all of them together:

(2*4*6*8*10*12*14*16*18*20)*(5*10*15*20*25)

4) Decomposing the numbers into prime factors:

(2*(2*2)*(2*3)*(2*2*2)*(2*5)*(2*2*3)*(2*7)*(2*2*2*2)*(2*3*3)*(2*2*5))*(5*(2*5)*(3*5)*(2*2*5)*(5*5))

5) Rearranging the factors in powers :

$(2^{18}* 3^4 * 5^2 *7) * (2^3 * 3^1 * 5^6)$

6) Grouping together powers of 2 and 5

$2^{21} * 5^8 *3^4 *7$

7) Joining together powers of 2 and 5

$10^8 *2^{13} * 3^5 *7$

8) Since one power of ten represents one zero at the end of the number.

9) Hence there will be 8 zeroes at the end of product