Question

The number 111111111 is a
(A). Prime number
(B). Composite number
(C). Divisible by $$\dfrac{10^{7}-1}{9}$$
(D). None of the above

Answer 1

Hint: In this question it is given whether the given number 111111111 is prime or composite or divisible by $$\dfrac{10^{7}-1}{9}$$. So for this we have to check for each option one by one. So for this we have to expand the given number in the form of summation. So for this we need to know a formula for Geometric progression, which states, if a be the first term and r be the common ratio of an G.P then the summation of first n terms, $$S_{n}=a\dfrac{r^{n}-1}{r-1}$$ ………(1)

Complete step-by-step solution:
Given number is 111111111.
Let $$S_{n}=111111111$$, which can be written as in the form,
$$S_{n}$$=1+10+100+1000+......+100000000
So in the above series the total number of terms are 9, and the first term of the above series is 1,i.e, a=1.
Now we are going to check whether the given ratio is common or not,
Let, $$r_{1}=\dfrac{\text{second term} }{\text{first term} } =\dfrac{10}{1} =10$$
$$r_{2}=\dfrac{\text{third term} }{\text{second term} } =\dfrac{100}{10} =10$$
$$r_{3}=\dfrac{\text{fourth term} \text{third term} }{\text{second term} } =\dfrac{1000}{100} =10$$
$$\therefore r_{1}=r_{2}=r_{3}$$
Hence the ratio is common , so r=10,
So we can say that the above series is in the form of G.P
Then the summation, (n=number of terms=9)
$$S_{9}=a\dfrac{r^{n}-1}{r-1}$$
  $$=1\times \dfrac{10^{9}-1}{10-1}$$
  $$=\dfrac{10^{9}-1}{9}$$
So we get, $$111111111=\dfrac{10^{9}-1}{9}$$
Therefore we can say that the given number is not divisible by $$\dfrac{10^{7}-1}{9}$$.
Which implies option C is wrong.
Now we are going to check whether the given number is prime or not.
Now if we add each and every digit of the given number then we get,
1+1+1+1+1+1+1+1+1=9, which is divisible by 3.
Therefore the given number is also divisible by 3.
So apart from 1 and the number itself there is another factor which is 3, so by this we can easily say that the given number is not a prime number, which implies composite number.
Hence the correct option is option B.

Note: To find whether the given number is divisible by 3 or not, we need to know that if the summation of the all digits of a number is divisible by 3 then the number itself is divisible by 3. Also while identifying a prime number, the basic definition you have to know, which states that if any number has only two factors one is 1 and another one is the number itself, then the number is prime and if a number has more than two factors then the number is called composite number.