Number Theory Solved Problems

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A Couple Of Simple Number Theory Problems And Solutions Many basic number theory problems relate to factoring.

Following are a couple of examples: Problem: You have a quantity of cookies.

The prime factorisation of the integers is a central point of study in number theory and can be visualised with this Ulam spiral variant.

Number theory seeks to understand the properties of integer systems like this, in spite of their apparent complexity.

and had a problem similar to the Pythagorean theorem before Pythagoras lived and much before Euclid (300BC).

Another early historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca.

Solution: 5281 is a prime number, so it cannot be subdivided into smaller equal groups. 5106 ends in an even number, so it must be divisible by 2.

In the case of 5751, the sum of its digits (5 7 5 1=18) is divisible by three, so 5751 must be divisible by 3.

Prove that if ^n-1$ is a Mersenne prime number, then \[N=2^(2^n-1)\] is a perfect number.

On the other hand, prove that every even perfect number $N$ can be written as $N=2^(2^n-1)$ for some Mersenne prime number ^n-1$.


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