Question

Write all factors of the following number 729.

Answer 1

Hint: To find the factors 729. At first we have to check the divisibility of 729 by prime numbers and pick out the prime factors. Then by obtaining all the prime factors we must combine them in all possible ways and their products will give the factors.

Complete step-by-step answer:
Before we solve the question, let us see what the meaning of factor of any number is.
A factor is a number that divides into another number exactly and without leaving a remainder, or we can say, a number ‘x’ is a factor of number ‘y’ if x divides y by giving remainder equals 0.
For example, let us find a factor of 10.
We know that, $10=1\times 2\times 5$ , which means 10 can be divided by 1, 2 and 5 leaving remainder equal to 0, also number itself is its own factor.
So, factors of 10 are 1, 2, 5 and 10.
Now, in question we are asked to find the factors of 729. Always remember that 729 is a perfect square that is $729={{(27)}^{2}}$ , then it will have all odd number of factors in count.
We know that all numbers have 1 and the number itself as common factors. So 1 and 729 must be the factors of 729.
Then by divisibility test, 729 are only divisible by only the prime number 3.
Doing prime factorization of 729, we get
$\begin{align}
  & 3\left| \!{\underline {\,
  \text{ 729} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \text{ 243} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \text{ 81} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \text{ 27} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \text{ 9} \,}} \right. \\
 & 3\left| \!{\underline {\,
  \text{ 3} \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  \text{ 1} \,}} \right. \\
\end{align}$
 Hence 729 can be expressed as \[729=3\times 3\times 3\times 3\times 3\times 3\]. Therefore these 6 3’s are combined in all possible ways and their products are given by,
\[\begin{align}
  & 1\times 3=3 \\
 & 3\times 3=9 \\
 & 3\times 3\times 3=27 \\
 & 3\times 3\times 3\times 3=81 \\
 & 3\times 3\times 3\times 3\times 3=243 \\
\end{align}\]
Hence we got that the factors of 729 are 1,3,9,27,81,243

Note: A prime number has only two factors, those are 1 and the number itself. Also, remember that a perfect square is a total odd number of factors in count. A number itself is its own factor. Try not to make any calculation mistakes while doing factorization of numbers.