Question

Explain why \[\left( {3 \times 5 \times 13 \times 46} \right) + 23\] is a composite number.

Answer 1

Hint: In the above question, we are given a number in the form of a product of four numbers and the sum of another number. The given number is written as \[\left( {3 \times 5 \times 13 \times 46} \right) + 23\] . We have to explain why it is a composite number. First, we need to determine if it is a composite number or not. In order to approach the solution, we need to rewrite the above number in a form such that by which we can determine if it is a composite number or not.

Complete step by step answer:
The given number is
\[ \Rightarrow \left( {3 \times 5 \times 13 \times 46} \right) + 23\] .
We have to explain why the above number is a composite number.
To determine that, first we need to check if it is even a composite number or not, and by the process of checking our answer we can obtain the reason of that why the given number is a composite number.
To check if \[\left( {3 \times 5 \times 13 \times 46} \right) + 23\] is a composite number or not, we need to rewrite the number in a way that it looks like a composite number, only if it is actually a composite number.
Now, since the number is \[\left( {3 \times 5 \times 13 \times 46} \right) + 23\] , we can notice that all the numbers are an odd number except \[46\] , which is in case actually a multiple of \[23\] .
So we can rewrite \[46\] as a multiple of \[23\] as \[46\] is two times \[23\] , hence we get
\[ \Rightarrow 46 = 2 \times 23\]
Now, replacing \[46\] by \[2 \times 23\] in the given number we can rewrite the number as,
\[ \Rightarrow \left( {3 \times 5 \times 13 \times 46} \right) + 23\]
Hence,
\[ \Rightarrow \left\{ {3 \times 5 \times 13 \times \left( {2 \times 23} \right)} \right\} + 23\]
Now, taking \[23\] as common from both parts, we can write,
\[ \Rightarrow 23\left\{ {\left( {3 \times 5 \times 13 \times 2} \right) + 1} \right\}\]
Or,
\[ \Rightarrow 23 \times \left\{ {\left( {3 \times 5 \times 13 \times 2} \right) + 1} \right\}\]
Therefore, the given number is nothing but actually the sum of the product of two numbers that are \[23\] and \[\left\{ {\left( {3 \times 5 \times 13 \times 2} \right) + 1} \right\}\] .
Hence, the number \[\left( {3 \times 5 \times 13 \times 46} \right) + 23\] is a composite number.

Note:
There are two types of numbers according to the classification based on the factorisation of a number. The number which has more than two factors, i.e. factors other than one and the number itself, such numbers are called as the composite numbers. Whereas, if a number has only two factors as one and the number itself, then such numbers are known as the prime numbers.