Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Euler's Remainder Theorem [#permalink]
04 Oct 2013, 22:47

2

This post received KUDOS

Euler's Remainder Theorem:

Euler’s theorem states that if p and n are coprime positive integers, then ———> P φ (n) = 1 (mod n), where Φn=n (1-1/a) (1-1/b)….

Before we move any further, let us understand what mod is. Mod is a way of expressing remainder of a number when it is divided by another number. Here φ (n) (Euler’s totient) is defined as all positive integers less than or equal to n that are coprime to n. (Co-prime numbers are those numbers that do not have any factor in common.)

For example ———-> 24=23 x 3

———-> Therefore, we get 24 x (1-1/2) (1-1/3) = 8

———-> which means that there are 8 numbers co-prime to 24.

They are 1, 5, 7, 11, 13, 17, 19, 23. Let us understand this theorem with an example:

Q.1) – Find the remainder of (7^100) / 66

Answer-

As you can see, 7 and 66 are co-prime to each other.

Therefore, Φ 66 = 66 x (1-1/2) x (1-1/3) x (1-1/11) = 20

Fermat's Little Theorem [#permalink]
04 Oct 2013, 22:57

2

This post received KUDOS

Fermat’s theorem is an extension of Euler’s theorem. If, in the above theorem, n is a prime number then, pn-1 =1(mod n)

Consider an example;

Q.2) Find remainder of 741 is divided by 41.

Here, 41 is a prime number.

Therefore, [7 40 x 7 /41] (By Fermat’s theorem)

which is equal to 7.

TB – Wieferich prime: is a prime number p such that p2 divides 2p − 1 – 1 relating with Fermat little theorem, Fermat’s little theorem implies that if p > 2 is prime, then 2p − 1 – 1 is always divisible by p....

Chinese Remainder Theorem [#permalink]
04 Oct 2013, 23:06

2

This post received KUDOS

Let’s understand this theorem with an example:

Q.5) – Rahul has certain number of cricket balls with him. If he divides them into 4 equal groups, 2 are left over. If he divides them into 7 equal groups, 6 are left over. If he divides them into 9 equal groups, 7 are left over. What is the smallest number of cricket balls could Rahul have?

Let N be the number of cricket balls.

N = 2(mod4) ————–> equation 1

N = 6(mod7) ————–> equation 2 &

N = 7(mod9) ————–> equation 3.

From N=2(mod4) we get, N=4a+2

Substituting this in equation 2, we get the following equation:

4a + 2 = 6(mod7)

Therefore, 4a = 4(mod7)

Hence, 2 x 4a = 2 x 4(mod7)

This gives us a = 1(mod7)

Hence a = 7b+1.

Plugging this back to N=4a+2, we get….

N = 28b + 6

Substituting this to equation 2;

28b + 6 = 7(mod9)

28b = 1(mod9)

Therefore, b=1(mod9)

Hence b = 9c + 1.

Substituting this back to equation N=28b+6;

N = 28(9c+1) + 6

N = 252c + 34

The smallest positive value of N is obtained by setting c=0.

It gives us N = 34

TB – All prime numbers greater than 3 can be expressed as 6K+1 or 6K-1, this is another important result. You would be using this result a lot when it comes to number system problems...

1. My favorite football team to infinity and beyond, the Dallas Cowboys, currently have the best record in the NFL and I’m literally riding on cloud 9 because...

I couldn’t help myself but stay impressed. young leader who can now basically speak Chinese and handle things alone (I’m Korean Canadian by the way, so...