Odd and even numbers

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In mathematics, an even number is an integer that is divisible by two. It is an integer that can be written in the form: 2k (ie, integerly divisible by 2), where k is an integer (even numbers are multiples of the number 2). Integers that are not even are called odd numbers (or minor numbers), and can be written as 2k+1.

The even numbers are:

{displaystyle mathrm {pares} ={;... -14, -12, -10, -8, -6,;-4,;-2,;0,;2,;4,;6,;8, 10, 12, 14,;...;}}

and the odd ones:

{displaystyle mathrm {impares} ={;...,; -15, -13, -11, -9, -7, -5,;-3,;-1,;1,;3,;5,;7, 9, 11, 13, 15,;...;}}

The parity of an integer refers to its attribute of being even or odd. Two numbers are comparatively "of equal parity" if, when divided by 2, the remainders are the same, for example: "2" and "4", or "3" and "7"; they are “of the same parity”. Conversely the numbers "23" and "44" they are “of different parity”.

This is complemented by an easy formula:

pair + pair = pair | even + odd = odd | odd + odd = even

Acknowledgment

If the numbering base used is an even number (for example, base 10 or base 8), an even number can be recognized if its last digit is also even. For example, the following number in base 10:

{352107706}_{10}

is even since its last digit: 6, is also even. The same happens with the following number in base 6:

{displaystyle {2145301354}_{3}={23211718}_{100}}

If the base of the numbering system is odd (3, 5, etc), the number will be even if the number of digits with an odd number is even, otherwise the number will be odd. For example, in base 3:

{120}_{3}={15}_{10}

is odd, since one is the only odd number, while:

{321}_{5}={86}_{10}

Since 3 and 1 are odd, there is an even number of odd digits and the number is even.

Zero parity

Main article: Zero parity

Zero is an even number, it meets the definition as well as all the properties of even numbers.

$I_{1}+I_{2}=2a+1+2b+1=2a+2b+2=2(a+b+1)=2n$
$P_{1}cdot P_{2}=2acdot 2b=2(2cdot acdot b)=2(c)=2n$
$P_{1}cdot I_{1}=2acdot (2b+1)=2acdot 2b+2a=2c+2a=2(c+a)=2n$
$I_{1}cdot I_{2}=(2a+1)cdot (2b+1)=2acdot 2b+2a+2b+1=2c+2a+2b+1=2(c+a+b)+1=2n+1$
The base power pairs are pairs and reciprocally if a power is pair its base is pair
The rest of the division of a pair number between a pair number is pair; nothing collids from the quotient that can have any parity.

Properties with respect to divisibility

Two consecutive integers have different parity.
Given three consecutive integers, two will be of the same parity and one of them will necessarily be parity other than the other two.

Special types of even numbers

The perfect numbers are pairs.
The factors of a natural different from 1 and 0 and the primeval numbers are pairs.
All primitive Pythagorean (i.e., all terna of integers ${displaystyle (alphabetagamma)}$ such that there are two positive integers among themselves ${displaystyle m,n}$ such that $n}" xmlns="http://www.w3.org/1998/Math/MathML">m▪n{displaystyle mpurn}n}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637039c4a193f33fee72ebfeb6cb003593696160" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;"/>$ and ${displaystyle alpha =2mn}$ , ${displaystyle beta =m^{2}-n^{2}}$ and ${displaystyle gamma =m^{2}+n^{2}}$ ) generates a congruent number (that is, an area value of a rectangle triangle of sides rational numbers) par.

Special types of odd numbers

The prime numbers, with the only saving of 2, which is pair. It is those natural numbers that do not have other divisors more than themselves and 1.
- The prime numbers of the form $4cdot n+1$ , with n any natural number, decompose in a single way in sum of two squares of integers. This was studied by Fermat and allows that cousin to be the hypotenuse of a diophantic rectangle or diophantine rectangle triangle. These last two words refer to triangles with positive integer sides in honor of Diofanto of Alexandria, who studied the problems in which to obtain whole solutions.
- The cousins of the way $4cdot n+3$ cannot be expressed as a sum of two whole squares, but yes as a difference of squares. The square root of the larger square, or minuendo of the difference, is equal to $2(n+1)$ where n is the same natural that appears in the expression of the prime number.

Deprecated definitions

In book 7 of Euclid's Elements (definitions 8 to 10), some classes of numbers are defined which, although in disuse today, have been repeatedly cited in historical books on math.

Number parly par, pariter par or proper pair "is the one measured by a number pair according to a number pair". It would therefore be the product of two pair numbers (all are multiples of 4).
Number even odd or odd pariter "is the one measured by a number pair according to an odd number", that is, the product of a number pair by an odd number.
Unforeseen number, impariter or proper odd "is the one measured by an odd number according to an odd number", that is, the product of two odd numbers.

Observations:

In these definitions, the 1 does not count as number, so the oddly odd numbers are exactly the composite odd numbers. These are the numbers used in the Sundaram shed to find prime numbers: a prime number will be every odd number (with the conspicuous exception of 2) that is not in the Sundaram shed.
Some numbers are considered both pairs and parly odds. For example, 24 is equal to 6 by 4, so it is parly pair; but it is also equal to 3 by 8, with what is parly odd.

Some fonts, such as Dorado counter. Speculative and Practical Arithmetic (1794) and the most recent, Mathematical Swarm, use another definition for even numbers: they are not the products of two pairs, but of which can only be expressed as the product of two pairs (excepting, of course, the product of themselves times one). According to this definition, the even numbers are exactly the powers of 2. Likewise, they define the evenly odd number as the multiple of a power of 2 by an odd number and introduce the concept, absent in Euclid's work, of odd number. even as a number that is twice an odd number. The definition of the odd odd number does not change.

The book Arithmetic and Algebraic Key uses the first definitions and explains the case of numbers that are simultaneously even and partially odd. This definition, moreover, is reinforced in proposition 32 of book 9 of the Elements, which explains as follows: «Each of the numbers (which is continuously) duplicated from a dyad is only a (number) partly even."

Even divisibility

Be the set of pairs $2mathbb {Z}$ = {0, 2, 4, 6, 8, 10,...2n... n any natural}.

Sean a b, c elements $2mathbb {Z}$ , it will be said that a日本語p b if there is c such that b = ac. It is also said that b is divisible parly; For example 8 Ș 16 because 16 = 2·8
Cousin $2mathbb {Z}$

element a is prime in 2Z if there is no element of 2Z that divides it.

For example, 6, 10, since there is no element of 2Z that divides it evenly.

The cousins of $2mathbb {Z}$ are the product of the odds by 2 only.

Number dividers

Outside of the even primes, the other numbers have more than two divisors

For the case of 24, it has as dividers 2, 4, 6, 12, and 8 is not divisor parly of 24.

Common dividers

48 and 32 have common divisors of 2, 4, 8, and 16 does not, because it does not evenly divide 48

Greatest Common Divisor Partly

The largest of the common dividers of two elements $2mathbb {Z}$ is called common divisor maximum (m.c.d.).

For example, m.c.d.(32.48) = 8

Algebra

Add, subtraction and multiplication of integers:
- par ± pair = pair
- par ± odd = odd
- odd ± odd = pair
- pair = pair
- pair = pair
- impar·impar = odd
The sum of natural numbers pairs is pair and fits the associative property, the set of the pair numbers is a semi commutative with the addition; if you admit 0 as natural, would be the aditive neutral element pair.
The set of the integers pairs with the addition is an abelian group, as they are fulfilled: the closure, associativeness, there is the neutral element to zero and for each pair there is its opposite.
The set of odd natural numbers with multiplication is a semi-associative group, with unity.

Power parity

Number $a$ It's even if and only if ${displaystyle a^{2}}$ It's a number pair. This property is used in the demonstration of the irrationality of ${displaystyle {sqrt {2}}}$ .
Number $a$ It's odd if and only if ${displaystyle a^{2}}$ is an odd number (consequence of the above).

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