Question

Which of the following number will completely divide $\left( {{3^{25}} + {3^{26}} + {3^{27}} + {3^{28}}} \right)$?
(A) $11$
(B) $16$
(C) $25$
(D) $30$

Answer 1

Hint: Here in this question, first we will common out the smallest term among them which are written in addition. Then, we will add all the terms which are in the bracket and then we can reduce some powers of three and then after rearranging some terms we will get to our answer.

Complete answer:
In the above question, we have to find the number which will completely divide the above number.
First, we will common out the smallest term from the terms which are in addition in the above question.
Therefore,
$ = {3^{25}}\left( {1 + {3^1} + {3^2} + {3^3}} \right)$
Now, simplify the terms which are in the bracket.
$ = {3^{25}}\left( {1 + 3 + 9 + 27} \right)$
Now, we will add all the terms in the bracket
$ = {3^{25}}\left( {40} \right)$
We can also write the above equation as
$ = {3^{25}} \times 4 \times 10$
Now we will reduce on power of three and multiply three in the above equation.
$ = {3^{24}} \times 3 \times 4 \times 10$
We can also write it as
$ = {3^{24}} \times 30 \times 4$
Therefore, it is clearly seen that the above value is divisible by $30$.

Therefore, the correct option is D

Note: If we keep on reducing the powers of three, then there will be no change in the answer. It remains same. The above quantity is also divisible by all the prime factors of $30$. The above quantity is also divisible by $4$but it is not given in the options.