Question

If the sum of the digits of the number $ N={{2000}^{11}}-2011 $ is S, then
A. S is a prime number
B. Sum of digits of S is 10
C. (S+1) is divisible by exactly 3 prime numbers
D. S is a composite number

Answer 1

Hint: we can solve this question by writing $ 2000 $ in the form of its factors and ‘2’ and ‘10’ as $ 2000=2\times 1000=2\times {{10}^{3}} $ and then solve it as the number will end in $ 3\times 11=33 $ zeros and then 2011 will be subtracted from it and we will get the value of ‘N’ resulting in the value of ‘S’. Then we can check all the options based on the value of ‘S’ and find out the right options.

Complete step-by-step answer:
Now, we know that $ 2000=2\times 1000=2\times {{10}^{3}} $
Therefore, $ N={{(2\times {{10}^{3}})}^{11}}-2011 $
 $ \begin{align}
  & \Rightarrow N={{2}^{11}}\times {{10}^{33}}-2011 \\
 & \Rightarrow N=2048\times {{10}^{33}}-2011 \\
\end{align} $
Now we know that $ {{10}^{33}}=100000....\left( 33zeros \right) $
Therefore, $ N=2048\times 10000....(33zeros)-2011 $
 $ \begin{align}
  & \Rightarrow N=20480000....(33zeros)-2011 \\
 & \Rightarrow N=20479999....\left( 29nines \right)7989 \\
\end{align} $
Therefore sum of the digits of ‘N’ will be given by:
 $ \begin{align}
  & S=2+0+4+7+9+9+9+....+9\left( 29nines \right)+7+9+8+9 \\
 & S=2+4+7+9\times 29+7+9+8+9 \\
 & S=46+261 \\
 & S=307 \\
\end{align} $
Therefore the value of $ S=307 $
Now we will check for all the options.

Checking for option (A):
As 307 does not have any factor except for 1 and itself, it is a prime number.
Therefore, option (A) is a correct option.

Checking for option (B):
Sum of digits of S $ =3+0+7=10 $
Thus, the sum of digits of S is ‘10’ as given in the option.
Therefore, option (B) is a correct option.

Checking for option (C):
 $ S+1=307+1=308 $
After prime factorisation of S+1, we will get all the prime numbers S+1 is divisible by and then we will count them and see if they are equal to 3 or not.
Now, we know that $ 308={{2}^{2}}\times 7\times 11 $
Thus, S+1 is divisible by 3 prime numbers:2,7 and 11
Therefore, option (C) is also a correct option.

Checking for option (D):
As we have already established while checking for option (A) that S is a prime number therefore it cannot be a composite number at the same time.
Hence, option (D) is not a correct option.

So, the correct answer is “Option A,B and C”.

Note: Calculation of S should be done very carefully. After calculating $ {{2000}^{11}} $ , care should be taken while subtracting $ 2011 $ from it. $ 2011 $ will appear downside of the last 4 zeroes and the rest of the zeros would be converted into ‘9’.