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

_________________

Register for free sessions

Number Properties | Algebra |Quant Workshop

Success Stories

Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant

Articles and Question to reach Q51 | Question of the week

Must Read Articles

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2

Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2

Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability

Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry

Algebra- Wavy line | Inequalities

Practice Questions

Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com