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.
Problems
- Two classes bought 737 textbooks. Each student received the same number of textbooks. How many students were there in both classes together?
- The dividend is 371, and the remainder is 30. Find the divisor and the corresponding quotient.
- List all divisors of the following numbers:
a) 3 × 5 × 5 × 11
b) 5 × 45
c) 1001
d) 256 - Find the smallest composite number that is not divisible by any number less than 10.
- How many natural divisors does the number have?
a) p^q
b) p^2 * q^3, where p and q are prime numbers. - Let a be an integer such that a+1 is divisible by 3. Prove that 4 + 7a is also divisible by 3.
- What is the remainder when dividing the following numbers by 7?
a) 4395
b) 17645
c) 1,781,003 - Let a and b be integers such that (3a + 7b) is divisible by 19. Prove that (41a + 83b) is also divisible by 19.
- Prove that a number of the form "abcabc" (i.e., a six-digit number formed by repeating a three-digit sequence) cannot be a perfect square.
- Find all natural numbers p such that both p and 5p + 1 are prime numbers.
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.