Determine the gcd of each of the following pairs of numbers, and write the gcd as a linear combination of the given numbers:

1. 480, 175;
2. 5621, 219;
3. 983675, 105120.

Show that, for all positive integers x, y, z,

Let a, b be distinct integers.

1. Show that a - b | a² - b² and determine the quotient.
2. Show that a - b | a³ - b³ and determine the quotient.
3. Prove that a - b divides aⁿ - bⁿ for all positive integers n.
4. Show that a + b | a^{2n + 1} + b^{2n + 1} for each positive integer n.
5. Find the prime factorization of 5¹⁰ - 2¹⁰.

Prove that there exist infinitely many primes of the form 4n + 3.

If c is a common multiple of a and b, then c is a multiple of lcm(a,b). Prove this.

For which positive integers n is n² - 1 prime?

Which integers of the form 13p + 1 (with p prime) are squares of integers?

Prove: If a and b are integers, not both zero, and c = gcd(a,b), then c = min{xa + yb | xa + yb > 0, x, y Z}.

Prove that the cube root of 17 is not a rational number.

Find the factorization of 2³² - 1.

For any positive integer n divide 10³ⁿ by 10ⁿ - 1 and find the remainder.

Three cogwheels with 24, 15 and 16 cogs, respectively, touch as shown in the picture.

• What is the smallest positive number of times you have to turn the left-hand cogwheel before the right-hand cogwheel is back in its original position?
• What is the smallest positive number of times you have to turn the left-hand cogwheel before all three wheels are back in their original position?

Prove that the square of an odd integer is again odd, where `odd' means `not divisible by 2' or, equivently, `leaving remainder 1 upon division by 2'. Show that the remainder of division by 4 of the square of an odd integer is 1. Does the last statement hold if we replace 4 by 8? And by 16?

Prove each of the following statements.

• If a | b and b | c then a | c.
• If a | b and c | d then ac | bd.

Let a be a rational number such that both 18a and 25a are integers. Show that a itself is an integer.

Find all integers x and y such that 32x + 10y = 6.

Find all primes p and q such that 4p + 7q = pq.