Question

How can you use prime factorization to determine if 856 is evenly divisible by 7?

Answer 1

Hint: To find if 856 is evenly divisible by 7 first we need to find the prime factors of 856. Then by comparing the prime factors of 856 we will reach the conclusion that it is divisible by 7 or not.

Complete step by step solution:
We have been given a number 856.
We have to find that if 856 is evenly divisible by 7 or not.
First we need to find the prime factors of 856 by using the prime factorization method.
The prime factors of 856 will be
$\begin{align}
  & \text{ }2\left| \!{\underline {\,
  856 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  428 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  214 \,}} \right. \\
 & 107\left| \!{\underline {\,
  107 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\Rightarrow 856=2\times 2\times 2\times 107$
Here we get the prime numbers 2 and 107 as the prime factors of 856.
We know that the number 856 is divided by 2 and 107 leaves the remainder zero; it means it is divisible by 2 and 107.
Now, let us check if 107 is divisible by 7 or not. Then we will get
$\Rightarrow 107\div 7$
$7\overset{15}{\overline{\left){\begin{align}
  & 107 \\
 & \underline{-7} \\
 & \text{ }37 \\
 & \underline{-35} \\
 & \text{ }2 \\
\end{align}}\right.}}$
Now, on dividing 107 by 7 it leaves the remainder 2, it means it is not evenly divisible by 7.
Hence 856 is not evenly divisible by 7.

Note: The point to be noted is that we have to find only the prime factors of 856. If not mentioned in the question we can also check the divisibility by using the long division method. If remainder is zero then the number is divisible else the number is not divisible.