dejavu619 wrote:

I was just fooling around with remainders and had a query.

Suppose I want to find the remainder when \(7^{30}\) is divided by 100. I know I can solve this using cyclicity, but why don't I get the right answer using the following method?

I can express \((7^{30})/100\) as \([(7^{15})/10]^{2}\). Now why can't I find the remainder of \((7^{15})/10\) and simply square it to obtain the desired answer?

(Answer is 49)

We know that any number can be written in terms of its Divisor, Quotient and Remainder.

Thus, we can write \(N = QD + R\) ......(i)

According to your logic, if the remainder when N divided by D is R, then the remainder when \(N^2\) is divided by \(D^2\) should be \(R^2\).

If you square equation (i), you will be able to see it clearly that your assumption is not correct.

\(N^2 = (QD + R)^2\)

\(N^2 = Q^2* D^2 + R^2 + 2 * Q * R * D\)

Keep in mind that

you are dividing \(N^2\) by \(D^2\) now..

\(Q^2 * D^2\) is diivisble by \(D^2\).

But what about \(2*Q * R * D\), do we know if this is perfectly divisble by \(D^2\)?

No, we don't!

Hence, it is wrong to make such assumption that if \(N/D\) give \(R\) as remainder then \(N^2/D^2\) will give \(R^2\) as the remainder or vice versa.

Regards,

Saquib

e-GMATQuant Expert

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