CMI2010-A-07

Question

For integer values of $n$, the expression $\frac{n(5n + 1)(10n + 1)}{6}$

• A. Is always divisible by $5$.
• B. Is always divisible by $3$.
• C. Is always an integer.
• D. None of the above

Answer 1

Let $X=\frac{n(5n+1)(10n+1)}{6}$

For $n=1,  X= \frac{6*11}{6}=11$

For $n=2,  X=\frac{2*11*21}{6}=77$

For $n=3,  X=\frac{3*16*31}{6}=248$

For $n=4,  X=\frac{4*21*41}{6}=574$

Here, we can see $X$ is not divisible by $3$ and $5$ but $X$ is always an Integer.

Hence, Option(C) Is always an integer.

Comments

if u consider for negative integer values then it also gives integer values ..so C is correct

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How can you generalize the solution by just checking for few values of n?

You can obviously eliminate the answer by negativity test but how you can say that answer would be C, not D?

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Yes, the answer does not prove it

Answer 2

Option A and B can be eliminated easily, if you take n = 2 you get value of the given expression 

let X=n(5n+1)(10n+1)/6

=2(5*2+1)(10*2+1)/6

=77   --- which is neither divisible by 5 nor divisible by 3, Hence option A and B eliminated.

Option C

Proof by Induction:

Hence option C is correct.