Question

Find the highest power of six dividing the number:
\[72\times 727\times 7272\times 72727\times 727272\times 7272727\times 72727272\times 727272727?\]

Answer 1

Hint: There are various rules that are applied on numbers and their powers to find the values in an easier way.
The numbers and their powers are connected by four arithmetic operations – addition, subtraction, multiplication and division.

Complete step by step answer:
The rules on base and power of numbers are as follows:
a^m × aⁿ = (a)^{m + n}
a^m ÷ aⁿ = (a)^{m - n}
(ab)^m = a^m × b^m
a^m × b^m = (ab)^m
${{a}^{0}}=1$
Bases and powers can be negative or positive. This indicates that both bases and powers belong to rational numbers as rational numbers include all types of integers, zero and both positive and negative fractions.
The rules related to base and powers help in calculating complex problems in very less time.
The given expression is written as:
Expression = \[72\times 727\times 7272\times 72727\times 727272\times 7272727\times 72727272\times 727272727\]
In this expression, the odd numbers that are the numbers carrying odd number $7$ at unit/ones place are not divisible by six.
The remaining even numbers are: \[72\times 7272\times 727272\times 72727272\]_.
The number $72$ has factors as $72=2\times 6\times 6=2\times {{(6)}^{2}}$, $6$ to the power $2$ divides $72$. Similarly, the number $72$ is written as: $7272=6\times 6\times 202=202\times {{(6)}^{2}}$. It is also divisible by $6$ to the power $2$.
Number \[727272\] can be written as: $727272=6\times 6\times 6\times 3,367=3,367\times {{(6)}^{3}}$. This indicates it is divisible by $6$ to the power $3$. \[72727272\] can be written as: $72727272=6\times 6\times 2,020,202=2,020,202\times {{(6)}^{2}}$. This indicates it is divisible by $6$ to the power $2$.
When summing all powers of six diving the above four numbers, we get:
${{6}^{2}}\times {{6}^{2}}\times {{6}^{3}}\times {{6}^{2}}={{6}^{2+2+3+2}}={{6}^{9}}$
This implies that the highest power of six that divides given expression is $9$.
Note: The power of a number denotes the number of times that number has to be multiplied with itself.
For instance, ${{\left( 2 \right)}^{3}}$ denotes that number $2$ must be multiplied $3$ times to get answer equal to $2\times 2\times 2$ equal to $8$ and similarly if $8$ is to be written as power of some number, it is written as ${{(2)}^{3}}$.