Question

Resolve the following into prime factors.
(i) 784
(ii) 12544

Answer 1

Hint: To find the prime factors, we will use the ladder method. In this method, we will write the number horizontally on the right within an L-shaped structure. On the left, we will write its prime factors that completely divide the number (starting from the lowest prime number). We will write the quotient below. Then, we will write the prime factors of this quotient on the left of it and the resultant quotient will be written below. We will do this procedure till the quotient becomes 1.

Complete step by step solution:
We have to write the prime factors of each of the given numbers. For this, we will be using the ladder method. In this method, we will write the number horizontally on the right and on the left, we will write its prime factors that completely divide the number (starting from the lowest prime number). We will write the quotient below. Then, we will write the prime factors of the quotient on the left of it and the corresponding quotient will be written below. We will do this procedure till the quotient becomes 1.
Now, let us find the prime factors of each of the given numbers.
(i) To find the prime factors of 754, we will use the ladder method described above.
$\begin{align}
  & 2\left| \!{\underline {\,
  784 \,}} \right. \\
 & 2\left| \!{\underline {\,
  392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  196 \,}} \right. \\
 & 2\left| \!{\underline {\,
  98 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\text{ }\text{ }1 \\
\end{align}$
The structure of the ladder method is shown above. We can see that 784 is completely divisible by 2 and the quotient is 392 which is written below 784. Then, we have divided 3942 by 2 (completely divisible) and the quotient is written below, which is 196. We have done this procedure till we obtained 1 at the end.
Therefore, we can write $784=2\times 2\times 2\times 2\times 7\times 7$ .
Hence, the prime factors of 784 are 2 and 7.
(ii) Let us find the prime factors of 12544.
\[\begin{align}
  & 2\left| \!{\underline {\,
  12544 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6272 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3136 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1568 \,}} \right. \\
 & 2\left| \!{\underline {\,
  784 \,}} \right. \\
 & 2\left| \!{\underline {\,
  392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  196 \,}} \right. \\
 & 2\left| \!{\underline {\,
  98 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\text{ }\text{ }1 \\
\end{align}\]
Therefore, we can write $12544=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 7\times 7$ .
Hence, the prime factors of 12544 are 2 and 7.

Note: Students must divide the number and write its factors carefully. They must only use prime numbers in the ladder method. Prime numbers are numbers that have only 2 factors: 1 and themselves. We can also use the factor tree method. In this method, we will write the number at the top and draw two branches. Below one branch will be its prime factor that completely divides the number and below the other branch, we will write its quotient. Now, we will draw two branches for this quotient and repeat the steps until we get two prime numbers in the two last branches.
We can write 784 as
\[\begin{align}
  & \begin{matrix}
   {} & {} & 784 & {} \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & \swarrow \searrow & {} \\
\end{matrix} \\
 & \begin{matrix}
   {} & \text{ }2 & {} & 394 \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   \text{ }\swarrow \searrow & {} \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   2 & \text{ }196 \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & \text{ }\swarrow \searrow \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & 2 & {} & 98 \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & {} & {} & \swarrow \searrow \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & {} & \text{ }2 & \text{ }\text{ }\text{ }49 \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & {} & \text{ } & \text{ }\swarrow \searrow \\
\end{matrix} \\
 & \begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   {} & {} & {} & {} \\
\end{matrix}\begin{matrix}
   \text{ }7 & {} & 7 \\
\end{matrix} \\
\end{align}\]
Similarly, we can do 12544.