Very First Question from CAT 2005

Hello Friends,

Today, while just flipping through a bunch of question papers, I came across this Quant question from CAT 2005. In fact, it was the very first question of the paper. IT goes like this :-

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If x= (16^3 + 17^3 + 18^3 + 19^3)

then what is the remainder when x is divided by 70.

a) 0

b) 1

c) 69

d) 35

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Here, at first sight, it looks somewhat solvable through remainder theorem.

But the question setters have definitely, tried to take advantage of the narrow thinking that many of the students develop. Once we are through with the "Remainder theorem ", topic in our quant preparation, we become somewhat bent towards using it, the moment we see the word remainder feature in a question.

However, here, the trick lies in understanding and capitalising on the fact that:-

(a^3 + b^3) will always be divisible by (a+b)

Since, (a^3 + b^3) = (a+b) (a^2 + b^2 - ab)

So, here we have

16^3 + 17^3 + 18^3 + 19^3

If we rearrange the expression as : -

(16^3 + 19^3 ) + (17^3 + 18^3)

16+19 = 17+18 = 35

therefore each of the sub-expressions in (16^3 + 19^3 ) + (17^3 + 18^3)

are divisible by 35.

So rewriting the expression as :-

35L + 35 M

Also worth noting is that:-

(16^3 + 19^3 ) has an odd unit digit (as their unit digits add to odd (6+9))

(17^3 + 18^3) has an odd unit digit (as their unit digits add to odd (3+2))

therefore we Now know that,

(16^3 + 19^3 ) + (17^3 + 18^3) is even as well as divisible by 35.

This is enough to conclude that (16^3 + 19^3 ) + (17^3 + 18^3)

is purely divisible by 70 = (35 * 2).

Hence the remainder is 0.

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Ans = (a)

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Till Next Time,

Good Bye and Happy Problem Solving,

Ashish