Number Theory

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

  1. Two classes bought 737 textbooks. Each student received the same number of textbooks. How many students were there in both classes together?

  2. The dividend is 371, and the remainder is 30. Find the divisor and the corresponding quotient.

  3. List all divisors of the following numbers:
    a) 3 × 5 × 5 × 11
    b) 5 × 45
    c) 1001
    d) 256

  4. Find the smallest composite number that is not divisible by any number less than 10.

  5. How many natural divisors does the number have?
    a) p^q
    b) p^2 * q^3, where p and q are prime numbers.

  6. Let a be an integer such that a+1 is divisible by 3. Prove that 4 + 7a is also divisible by 3.

  7. What is the remainder when dividing the following numbers by 7?
    a) 4395
    b) 17645
    c) 1,781,003

  8. Let a and b be integers such that (3a + 7b) is divisible by 19. Prove that (41a + 83b) is also divisible by 19.

  9. 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.

  10. Find all natural numbers p such that both p and 5p + 1 are prime numbers.

Interesting Problems 

  1. What is the remainder when dividing 9^2015 + 7^2015 - 2^2015 by 8?

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

  3. Prove that (abc - cba) is divisible by 99.

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