6. Elements of number theory

6.1. Fundamental Concepts

If w is a natural number then for any integer number a there exists a pair of integer numbers ^ and r such that

a = w - ^ + r, 0 < r < w.

A number ^ is called a quotient, and a number r is called a remainder. If a can be divided by m without remainder then we denote w a.

The least common multiple (LCM) of two (or more) nonzero whole numbers is the smallest nonzero whole number that is the multiple of each all of the numbers. LCM of a and A is written [a,A .

E.awT'/e d.7.7. Find [24,36 .

Express the numbers 24 and 36 in their prime factor exponential form: 24 = 2³-3, 36 = 2²-3². ^^e^ 2; The LCM will be the number 2³ - 3² .

The greatest common factor (GCF) of two (or more) nonzero whole numbers is the largest nonzero whole number that is a factor of both (all) of the numbers. GCF of a and A is written (a, A).

If (a₁,a₂,...,a^) = 1 then numbersa₁,a₂,...,a^ are called mutually prime numbers.

T^eorew. If a = A - ^ + r, then (a,A)=(A, r).

Froo/. If ^ A and ^ r then ^ a. If ^ a and ^ A then ^ r. Therefore a set of

divisors of A and r coincides with a set of divisors of a and A. Hence their greatest common factors are equal.

6.2. Euqludean Algorithm

Let a and A be positive integers, and a > A. We can find

a = A - + r1, 0 < r₁ < w₁; A = r₁ - + 0 < r₂ < r₁;

^r1 = ^r2 ^- ^3 + ^rз^{, 0} < ^r3 < ^r2;

^r"-2 = ^r"-1^- + ^r"^{, 0} < ^r" < ^r"-1;

^r"-1 = ^r" - . As a result we have

^{(a, A )} = ^{(A, r}1 ⁾=^(rl^{, r}2 ⁾= ... = ⁽r,-l^{, r}" ⁾= ^r" .

Exaw^/e d.2.7. Find (525,231).

231

2

525 462

63

3

231 189

	63	42
	42	1
42	21
42	2

0

Therefore (525,231)= 21.

6.3. Congruences and Their Properties

De/'M'Y'oM. Let w be a positive integer. We say that a is congruent to A modulo written a = ^(mod w) if w divides the difference a - ^. The integer w is called the modulus. For example

1. 87 = 23(mod 4) since 4 divides 87 - 23 = 64,

2. 67 = 1(mod6) since 4 divides 67 - 1 = 66,

3. 72 = -5(mod7) since 7 divides 72 -(-5) = 77,

4. 27^ 8(mod 9) since 9 does not divide 27 - 8 = 19.

^ewar^; Suppose w is positive, and a is any integer then there exist integers ^ and r with 0 < r < w such that a = w^ + r. Hence

w^ = a - r or w (a - r) or a = r(mod w).

Accordingly:

1. Any integer a is congruent modulo w to a unique integer in the set {0,1,2,...,w -1}. The uniqueness comes ^om the fact that w cannot divide the difference of two such integers.

2. Any two integers a and ^ are congruent modulo w if and only if they have the same remainder when divided by w.

Now we consider some properties of congruences.

1. Suppose a = c(mod w) and ^ = ^ (mod w). Then a + ^ = c + ^ (mod w) and a - ^ = c - ^(mod w).

Let a = ^ + ^w, c = ^ + /w, then a + c = ^ + ^ + + /)w or a + c = ^ + ^(mod w); a - c = ^ - ^ + + + ^/w)= ^^ + wM.

2. Both sides of a congruence and modulus it is possible to divide by some common divisor.

Let h = ^(mod w); h = h^^, ^ = ^^^, w = w^^, then h^^ = ^^^ + ^w^^. Hence h^ = ^^ + ^w^ and h^ = ^^ (mod w^).

3. Both sides of a congruence we can divide by their common divisor if the latter and the modulus of the congruence are mutually prime.

Let h = ^(mod w); h = h^^,^h = ^^^, (w,^) = 1, then (h^ - ^^ = ^w. Since (w,^) = 1,

then w (h^ - ^^) and h^ = ^^ (mod w^).

4. Ifh = ^(mod w), then (h, ^ ) = (^, w).

Really, if h = ^(mod w), then h = ^ + /w and (h,^) = w).

E.hwp/e d.3.7. Observe that 2 = 8(mod6) and 5 = 41(mod6). Then:

1. 2 + 5 = 8 + 41(mod6) or 7 = 8 + 49(mod6);

2. 2 - 5 = 8 - 41(mod6) or 10 = 328(mod6).