Question

Explain why \[5 \times 7 \times 11 + 11\] is a composite number .

Answer 1

Hint: In this question, we need to explain why \[5 \times 7 \times 11 + 11\] is a composite number . First, let us know what the composite number is. A composite number is nothing but a positive integer that can be formed by multiplying two smaller positive integers. First, we need to solve the given expression \[5 \times 7 \times 11 + 11\] and find it’s value. Then we need to find that the number has any other factors other than one and itself . If it has any other factors then the number is said to be a composite number.

Complete step-by-step answer:
Given, \[5 \times 7 \times 11 + 11\]
Here we need to explain why \[5 \times 7 \times 11 + 11\] is a composite number .
First let us simplify the given expression.
\[\Rightarrow \ 5 \times 7 \times 11 + 11\]
On taking \[11\] as common,
We get,
\[11(5 \times 7 \times 1 + 1)\]
On simplifying,
We get,
\[\Rightarrow \ 11 \times (35 + 1)\]
On further simplifying,
We get,
\[\Rightarrow \ 11 \times 36\]
Now on using prime factorisation method, we can find the factors of \[36\] ,
That is \[36 = 2 \times 2 \times 3 \times 3\]
Thus we get,
\[\Rightarrow \ 11 \times 2 \times 2 \times 3 \times 3\]
\[\Rightarrow \ 396\]
Thus the given number is completely divisible by \[11,3,2\]
Since the expression \[5 \times 7 \times 11 + 11\] can be expressed as a product of prime factors , it is a composite number.
Final answer :
The expression \[5 \times 7 \times 11 + 11\] is a composite number.

Note: In order to solve these questions , we should have a strong grip over composite numbers. We should keep in mind that any two numbers which have more than two factors then it is said to be a composite number, else it will be a prime number. We can also directly check whether any number can be taken as common from the whole number then we can directly conclude that as a composite number for example in the above solution we have taken \[11\] common from the given numbers so they are composite numbers.