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.
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