## Category Archives: RMO Number Theory

### How to solve equations: Dr. Vicky Neale: useful for Pre-RMO or even RMO training

Dr. Neale simply beautifully nudges, gently encourages mathematics olympiad students to learn to think further on their own…

### Number theory has numerous uses

One of the fun ways to get started in mathematics at an early age s via number theory. It does not require deep, esoteric knowledge of concepts of mathematics to get started, but as you explore and experiment, you will learn a lot and also you will have a ball of time writing programs in basic number theory. One of the best references I have come across is “A Friendly Introduction to Number Theory” by Dr. Joseph Silverman. It is available on Amazon India.

Well, number theory is not just pure math; as we all know, it is the very core of cryptography and security in a world transforming itself to a totally digital commerce amongst other rapid changes. Witness, for example, the current intense debate about opening up an iPhone (Apple vs. FBI) and some time back, there was the problem with AES Encrypted Blackberry messaging services in India.

Number theory is also used in Digital Signal Processing, the way to filter out unwanted “noise” from an information signal or “communications signal.” Digital Signal Processing is at the heart of modem technology without which we would not be able to have any real computer networks.

There was a time when, as G H Hardy had claimed that number theory is the purest of all sciences as it is untouched by human desire. Not any more !!!

Can you imagine a world without numbers ?? That reminds me of a famous quote: “God created the natural numbers, all the rest is man-made.” (Kronecker).

More later,

Nalin Pithwa

### Fun with Chinese Remainder Theorem

A farmer is on the way to market to sell eggs when a meteorite hits his truck and destroys all of his produce. In order to file an insurance claim, he needs to know how many eggs were broken. He knows that when he counted the eggs by 2’s, there was 1 left over, when he counted the eggs by 3’s, there was 1 left over, when he counted the eggs by 4’s, there was 1 left over,

when he counted the eggs by 5’s, there was 1 left over, and when he counted them by 6’s, there was one left over, but when he counted them by 7’s, there was none left over. What is the smallest number of eggs that were in the truck?

More fun later…

Nalin Pithwa

### Chinese Remainder Theorem — some history — RMO

The first recorded instance of the Chinese Remainder Theorem appears in a Chinese mathematical work from the late third or early fourth century. Somewhat surprisingly, it deals with the harder problem of three simultaneous congruences:

We have a number of things, but we do not know how  many. If we count them by threes, we have two left over. If we count them by fives, we have three left over. If we count them by sevens, we have two left over. How many things are there?

Sun Tzu Suan Ching (Master Sun’s Mathematical Manual)

Circa AD 300, volume 3, problem 26.

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