Question

19. The largest positive integer n for which n^200 < 6^300 is
(1) 12
(2) 13
(3) 17
(4) 14​

Answer 1

Given condition,

$n^{200} < 6^{300}$

Now, lets modify the given condition

$(n^2)^{100} < (6^3)^{100}$

applying($\sqrt[100]{}$) on both sides of the condition, we get

$\sqrt[100]{(n^2)^{100}} < \sqrt[100]{(6^3)^{100}} \\ \\n^2 < 6^3$

Then from the given options we have to verify the above condition one by one.

1)

$12^2 < 6^3\\144 < 216$  

the condition is satisfied that $144 < 216$

2)

$13^2 < 6^3\\169 < 216$

here also the condition is satisfied

3)

$17^2 < 6^3\\289 < 216$

The condition is not satisfied since, $289 > 216$

4)

$14^2 < 6^3\\196 < 216$

Here, the condition is satisfied as $196 < 216$

Now, from the above verification we know that options (1),(2), and (4) getting satisfied.

Note: But, in the question it is asked the 'Largest positive integer' which is $14$ from the satisfied options.

∴ option $(4). 14$ is the correct answer.

#SPJ2