Question

Explain why \[17\times 5\times 11\times 3\times 2+2\times 11\] is a composite number?​

Answer 1

Hint: If the given expression or any number is divisible by other numbers, then we can say that the given expression or that number is a composite number. Composite number is a positive integer that has at least one divisor other than one and itself. Thus, we have to check whether the given expression is divisible by other numbers or not. We will first compute the value of the given expression and then perform prime factorisation to get the factors of the number. After which, we can check if it satisfies the requirements of a composite number or not.

Complete step by step answer:
The given expression is,
\[17\times 5\times 11\times 3\times 2+2\times 11\]
We have to find the value first. So, we will perform multiplication first and then add them.
Therefore, we can solve this expression as follows,
\[\begin{align}
  & =5610+22 \\
 & =5632 \\
\end{align}\]
Now, we have to check if this is a composite number or not. So, this can be done by performing the prime factorisation of the number 5632.
Hence, the prime factorisation of 5632 is as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  5632 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2816 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1408 \,}} \right. \\
 & 2\left| \!{\underline {\,
  704 \,}} \right. \\
 & 2\left| \!{\underline {\,
  352 \,}} \right. \\
 & 2\left| \!{\underline {\,
  176 \,}} \right. \\
 & 2\left| \!{\underline {\,
  88 \,}} \right. \\
 & 2\left| \!{\underline {\,
  44 \,}} \right. \\
 & 2\left| \!{\underline {\,
  22 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & 1 \\
\end{align}$
Therefore, pairing the factors, we have \[5632={{2}^{9}}\times 11\] .
It means that 5632 is divisible by 2, 11 and there can be others factors too.

So, it has more than two factors. Hence, it is a composite number.
Therefore, we have proved that \[17\times 5\times 11\times 3\times 2+2\times 11\] is a composite number.

Note: Now in this expression, we have the common factors before and after positive signs as 2 and 11. Therefore, if we take these out from the terms, probably 2 and 11 are the prime factors. It is a quick check and the simplest method to get a solution to this type of problem. So, we will have the expression as
\[\begin{align}
  & 17\times 5\times 11\times 3\times 2+2\times 11 \\
 & \Rightarrow \left( 2\times 11 \right)\left[ 17\times 5\times 3+1 \right] \\
\end{align}\]
If students do not understand this, then they can even conclude their answer after obtaining 5632. Since it ends with 2, it is an even number. We know that all even numbers except 2 are composite numbers. So, this is another way of concluding the answer.