Number Theory

Introduction to Number Theory

Modular arithmetic is an important tool in number theory. When dividing a number $a$ by $b$, we obtain a quotient and a remainder, expressed as:

$a = bq + r$

where $0 \le r < b$. Instead of writing out full division calculations, mathematicians use modular notation:

$a \equiv r \pmod{b}$

Divisibility and Prime Factorization

To determine how many divisors a number has, we use its prime factorization. If a number $N$ has the form:

$N = p^a \times q^b \times r^c$

where $p, q, r$ are prime numbers, then the total number of divisors is given by:

$(a+1)(b+1)(c+1)$

Interesting Problems

What is the remainder when dividing $9^{2015} + 7^{2015} - 2^{2015}$ by 8?

The number $A$ has 5 divisors, and the number $B$ has 7 divisors. Can the product $AB$ have exactly 10 divisors?

Prove that $abc - cba$ is divisible by 99.