Question

if n=2^3x3^4x7x15^6, then find the number of consecutive zeros in natural number n

Answer 1

n = 2³  *  3⁴  *  7  *  15⁶
   = 2³ * 3⁴ * 7 * 3⁶ * 5⁶

we get 0, when we multiply a 2 with a 5.

So there are three 2s, six 5s    and hence  we get 3 consecutive zeros in the value of n.

Answer 2

Answer:

The number of consecutive zeroes in the given expression n will be three.

Solution:

To find the number of consecutive zeroes we need to separate out number of 2’s and 5’s possible in the calculation. This is because, equal number of 2’s and 5’s only lead to the formation of 10. Thus,  

Given that

$\begin{aligned} \mathrm { n } & = 2 ^ { 3 } \times 3 ^ { 4 } \times 7 \times 15 ^ { 6 } \\\\ & = 2 ^ { 3 } \times 3 ^ { 4 } \times 7 \times ( 3 \times 5 ) ^ { 6 } \\\\ & = 2 ^ { 3 } \times 3 ^ { 4 } \times 7 \times 3 ^ { 6 } \times 5 ^ { 3 } \times 5 ^ { 3 } \\\\ & = \left( 2 ^ { 3 } \times 5 ^ { 3 } \right) \times 3 ^ { 10 } \times 5 ^ { 3 } \times 7 \\\\ & = 3 ^ { 10 } \times 5 ^ { 3 } \times 7 \times ( 10 ) ^ { 3 } \\\\ & = 3 ^ { 10 } \times 5 ^ { 3 } \times 7 \times 1000 \end{aligned}$

There should be 3 consecutive zeros in the given expression n.