Question

How many zeroes are there at the end of the following product?

1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Options:- (a) 10 (b) 12 (c) 14 (d) 15​

Answer 1

1×5×10×15×20×25×30×35=393750000   (1)

40×45×50×55×60×=297000000               (2)

 (1) ×     (2)                     =116943750000000000

so there will be 10 zeros at the end of the product,

so the correct option will be:-

option (a) 10

Answer 2

Given:

To find the last number of zeroes in 1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60

Solution:

Understand the fact that only a pair of $\[2 \times 5\]$ will generate a zero,

There factorize further and the pair of $\[2 \times 5\]$

$\[\begin{array}{l}1 = 1\\5 = 5\\10 = 5 \times 2\\20 = 5 \times 2 \times 2\\25 = 5 \times 5\\30 = 5 \times 2 \times 3\\35 = 5 \times 7\\40 = 5 \times 2 \times 2 \times 2\\45 = 5 \times 9\\50 = 5 \times 5 \times 2\\60 = 5 \times 2 \times 2 \times 3\end{array}\]$

Notice that ,

total number of 2 = 10

Total number of 5 = 12

Therefore the total number of pairs of $\[2 \times 5\]$ is 10

Hence the total zeroes  at the end of the given product is 10.