Divisibility by 4 of a number which is written in the decimal system can be tested as follows: the number is divisible by 4 if and only if the number formed by the two last digits is divisible by 4. Prove this statement.

Formulate an 8-test (i.e., a test for deciding divisibility by 8) for numbers in the decimal system.

How does one decide divisibility by 8 for a binary number?

Formulate a test and prove its correctness for divisibility by a - 1 in the a-ary system.

The decimal representation of a number is abcabc. Here a, b and c are elements from {0, 1, ..., 9}. Prove that 7, 11 and 13 are divisors of abcabc.

Prove that n⁴ + n² + 1 is divisible by 3 if n > 0 is not divisible by 3.

Is the converse of Fermat's little theorem, `if x^{p - 1} = 1 mod p for all x, not equal to 0 mod p, then p is a prime' also true?

Prove the following statements:

• 13 | (10⁶ - 1).
• 17 | (10⁸ + 1).
• If n 0 mod 5, then n⁴ + 64 is not prime.
• The number 2¹⁰⁰⁰ + 5 is not prime. [Hint : work modulo 3.]
• For every n >0, 3 is a divisor of 2²ⁿ - 1.

Determine the multiplicative inverses of the given elements or show that this inverse does not exist.

• 3 Z/37Z.
• 4 Z/14Z.

Solve each of the following equations:

• 2x = 37 mod 21.
• 5x = 15 mod 25.
• 3x = 7 mod 18.

Fermat conjectured that numbers of the form 2²ⁿ + 1 are prime. For n = 5 this conjecture does not hold. Prove with the help of the following observations that 641 | 2²⁵ + 1.

• 641 = 2⁹ + 2⁷ + 1 and so 2⁷ (2² + 1) = -1 mod 641.
• 2⁴ = - 5⁴ mod 641.

Prove: if p is prime and 0 < k < p, then p divides the binomial coefficient p choose k. In addition show that for all x, y Z/pZ the equality

holds.

What are the invertible elements of Z/4Z, Z/6Z and Z/12Z? Let p be a prime. What are the invertible elements of Z/p²Z?

Prove that there are no integers x, y, z satisfying x³ + y³ + z³ = 5.

Make a list of the positive values less than 11 that this expression takes.

• Write 377 in the 2-ary, 8-ary and the 16-ary system.
• Write 301 in the 7-ary system.
• The number a is written in the 7-ary system as 3456142. Is this number divisible by 6? Hint: how can you test divisibility by 9 in the decimal system?
• Show that it is easy to rewrite a number given in the binary system into the hexadecimal (16-ary) system (and vice versa).

The Toy car company produces a car every 11 minutes (the `Corollarium'). The company starts on the midnight from Sunday on Monday. The director wants to celebrate a multiple of thousand of produced Corollariums on the moment that this particular car leaves the plant, but only if this happens on Friday afternoon between 3 and 5 o'clock. How many days after the start, and at what time, is the first opportunity?

Let n N.

• Show that the decimal representation of 1/n is finite if and only if the prime divisors of n are 2 and 5.
• Show that the decimal representation of 1/n is periodic from a certain point. (For example 1/7 = 0.14285714285714285714 ... has period of length 6 and 1/30 = 0. 033333333 ... has a period of length 1.)

Solve the following system of equations:

2x = 37 mod 5 and 3x = 48 mod 7.

Solve the following system of equations:

x + y = 6 mod 11 and 2x - y = 8 mod 11.

Let c Z/nZ have a multiplicative inverse. Prove that a = b mod n if and only if ca = cb mod n.

Consider the RSA cryptosystem with modulus 2623 and with encoding number v = 37.

If we use the labelling a = 01, b = 02, ..., y = 25, z = 26 and if a space is represented by 00, then decode the following text, where in each group of four figures a pair of these symbols is encoded: