$$'n'$$ is selected from the set $$\left\{ {1,2,3,.....,100} \right\}$$ and the number $$2^n + 3^n + 5^n$$ is formed. Total number of ways of selecting $$'n'$$ so that the formed number is divisible by 4, is equal to
A.
50
B.
49
C.
48
D.
None of these
Answer :
49
Solution :
If $$n$$ is odd, $${3^n} = 4{\lambda _1} - 1,{5^n} = 4{\lambda _2} + 1$$
$$ \Rightarrow {2^n} + {3^n} + {5^n}$$ is not divisible by 4, as $${2^n} + {3^n} + {5^n}$$ will be in the form of $$4\lambda + 2.$$
Thus, total number of ways of selecting $$'n'$$ is equal to 49.
Releted MCQ Question on Algebra >> Permutation and Combination
Releted Question 1
$$^n{C_{r - 1}} = 36,{\,^n}{C_r} = 84$$ and $$^n{C_{r + 1}} = 126,$$ then $$r$$ is:
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 ; and then the men select the chairs from amongst the remaining. The number of possible arrangements is