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Try this beautiful problem on Calculus, useful for ISI B.Stat Entrance.

## Problem on Calculus | ISI B.Stat Entrance | Problem 696

If k is an integer such that lim \(\{{cos}^n(k\pi/4) – {cos}^n(k\pi/6)\} = 0\),

then

- (a) k is divisible neither by 4 nor by 6
- (b) k must be divisible by 12, but not necessarily by 24
- (c) k must be divisible by 24
- (d) either k is divisible by 24 or k is divisible neither by 4 not by 6

**Key Concepts**

Calculus

Limit

Trigonometry

## Check the Answer

But try the problem first…

Answer: (d)

Source

Suggested Reading

TOMATO, Problem 694

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

There are four options ,at first we have to check each options…..

If k is divisible by 24 then cos(kπ/4) = cos(kπ/6) = 1

\(\Rightarrow\) The limit exists and equal to RHS i.e. 0

If k is not divisible by 4 or 6 then cos(kπ/4), cos(kπ/6) both <1

Can you now finish the problem ……….

Final Step

Therefore ,

lim cosn(kπ/4), cosn(kπ/6) = 0. so we may say that

\(\Rightarrow \)The equation holds.

## Other useful links

- https://www.cheenta.com/divisibility-problem-from-amc-10a-2003-problem-25/
- https://www.youtube.com/watch?v=XOrePzJWFiE

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