Question

Find the smallest natural number $m > 90$ for which \[n = \underbrace {1111......1}_{m - times}\] is not a prime number. Hence find the value of \[m - 87\].

Answer 1

Hint: First of all, take 91 as the first number natural number greater than 90. We will observe that the number comes out to prime. Next, consider the number 92. Apply the divisibility rule of 11 on the new number formed. If the number is divisible by 11, then it will not be prime and hence the value of $m$. Substitute the value of $m$ in the expression \[m - 87\].

Complete step by step Answer:

Let us consider natural numbers greater than 90.
Let $m = 91$
Then, we have \[n = \underbrace {1111......1}_{91 - times}\], which is a prime number because it has only two factors.
Similarly, let us consider the natural number greater than 91, which is 92.
Now, we have to find if \[n = \underbrace {1111......1}_{92 - times}\] is a prime number.
There is an even number of terms in the above digit.
And, the digits in the number are the same, that is all are 1’s.
Then, the sum of the even places is equal to the sum of the odd places.
Therefore, the difference of the sum of even places and the sum of odd places will be 0.
And, we know that if the difference of the sum of even places and the sum of odd places is 0, then the number is divisible by 11.
This implies that 11 is a factor of the number \[n = \underbrace {1111......1}_{92 - times}\]
Hence, the number \[n = \underbrace {1111......1}_{92 - times}\] is not a prime number.
Therefore, the value of $m$ is 92.
But, we want to calculate the value of \[m - 87\], therefore, substitute the value of $m = 92$ in the given expression.
Thus, we get,
$92 - 87 = 5$
Hence, the value of \[m - 87\] is 5.

Note: Students must know the divisibility rules for these types of questions. A number with more than 2 factors is known as a composite number. Also, do not forget to calculate the value of \[m - 87\] after finding the value of $m$.