Divisibility – SpeedyLook encyclopedia

In mathematics, specifically in arithmetic, it is said that an integer b is divisible Among another integer a (NO NULL) If there is another integer c such that:

b = a ⋅ c {displaystyle b=acdot c}

. This is equivalent to saying that the rest of the Euclidean division is zero or symbolically

b – a ⋅ c = 0 {displaystyle b-acdot c=0}

.

It is usually expressed in the form

a ∣ b {displaystyle amid b}

, to read: « a divide a b “, O ” a is A divisor of b ” or also ” b is multiple of a ». ^{[ first ]} For example, 6 is divisible between 3, since 3 × 2 = 6; But 6 is not divisible between 4, because there is no whole c such that 4 × c = 6; That is to say that the rest of the Euclidean (whole) division of 6 between 4 is not zero.

Any natural number ^{[ 2 ]} It is divisible between 1 and with each other. The numbers greater than 1 that do not admit more than these two divisors are called prime numbers. Those who admit more than two divisors are called compound numbers.

Definition [ To edit ]

The integer

a {displaystyle a}

is divisible between the entire number

b ≠ 0 {displaystyle bneq 0}

(Or what is the same, b divide a a ) If there is a number

q {displaystyle q}

whole that

a = b ⋅ q {displaystyle a=bcdot q}

.

This fact is called divisibility of the entire number

a {displaystyle a}

For the entire number

b {displaystyle b}

And it is denoted by

b | a {displaystyle b|a}

; That is nothing other than an affirmation between whole numbers, which, in a specific context, can be true or not. ^{[ 3 ]} For example

3 | twelfth {Displaystyle 3 | 12}

it’s true; however,

3 | 17 {Displaystyle 3 | 17}

It is not true. Yeah

b {displaystyle b}

It is not a divisor of

a {displaystyle a}

is written

b ∤ a {displaystyle bnmid a}

. Note that

0 ∤ a {displaystyle 0nmid a}

for all

a {displaystyle a}

different from zero, then

a ≠ 0 = k ⋅ 0 {displaystyle aneq 0=kcdot 0}

for all

k {displaystyle k}

whole.

Factor the propio divisor [ To edit ]

It is called factor the propio divisor entire number n , to another whole number that is a divisor of n , but different from 1 and n . Divisores 1 and n They are called improper .

For example, the divisors of 28 are 2, 4, 7 and 14. When they are taken into account negative, an own divisor is the one whose absolute value is lower than the given number. In this case, their own divisors would be -14, -7, -4, -2, 2, 4, 7, 14.

Special cases: 1 and -1 are trivial factors of all integers, and each integer is a divisor of 0. The numbers divisible by 2 are called pairs and those that are not called odd.

And d It is a divisor of a And the only divisor that admits D is 1 and himself, is called a cousin divisor of a . In fact it is a prime number. The 1 is the only whole that has a single positive divisor.

Properties [ To edit ]

Sean

a , b , c ∈ WITH {Displaystyle A, B, CIN MATHBB {z}}

, that is to say

a {displaystyle a}

,

b {displaystyle b}

and

c {displaystyle c}

They are whole numbers. Basic properties are given:

• And
  a ≠ 0 {displaystyle aneq 0}
  so a ∣ a {displaystyle amid a}
  (Reflexive property).
• and
  a ∣ b {displaystyle amid b}
  and b ∣ a {displaystyle bmid a}
  so | a | = | b | {displaystyle |a|=|b|}
  . They are equal or one is the opposite of the other.
• When
  a ∣ b {displaystyle amid b}
  and b ∣ c {displaystyle bmid c}
  , so a ∣ c {displaystyle amid c}
  (Transitive property).
• And
  a ∣ b {displaystyle amid b}
  and b ≠ 0 {displaystyle bneq 0}
  , so | a | ≤ | b | {displaystyle |a|leq |b|}
  .
• 
  a ∣ b {displaystyle amid b}
  and a ∣ c {displaystyle amid c}
  , it implies a ∣ b b + c c ∀ b , c ∈ WITH {displaystyle amid beta b+gamma c forall beta ,gamma in mathbb {Z} }
  . Diviser of the linear combination.
• 
  a ∣ b {displaystyle amid b}
  and a ∣ c {displaystyle amid c}
  , it implies a ∣ th b k+ K c j ∀ th , K , k , j ∈ WITH {Displaystyle Amd Theta B^{K}+Cappa C^{J} For thetall Theta ,Kppa ,K,Jin Mathbb {Z} }
  . Divisor of the linear combination of powers. ^{[ 4 ]} ​
• And
  a ∣ b {displaystyle amid b}
  and a ∣ b ± c {displaystyle amid bpm c}
  , so a ∣ c {displaystyle amid c}
  .
• Of
  a ∣ b {displaystyle amid b}
  and a ≠ 0 {displaystyle aneq 0}
  , is deduced ba∣ b {displaystyle {frac {b}{a}}mid b}
  . Conjugated divisors.
• For
  c ≠ 0 {displaystyle cneq 0}
  , a ∣ b {displaystyle amid b}
  If and only if a c ∣ b c {displaystyle acmid bc}
  .
• And
  a ∣ b c {displaystyle amid bc}
  and mcd ⁡ ( a , b ) = first {displaystyle operatorname {mcd} (a,b)=1}
  , so a ∣ c {displaystyle amid c}
  .
• When
  mcd ⁡ ( a , b ) = first {displaystyle operatorname {mcd} (a,b)=1}
  and c {displaystyle c}
  fulfills that a ∣ c {displaystyle amid c}
  and b ∣ c {displaystyle bmid c}
  , so a b ∣ c {displaystyle abmid c}
  .
• 
  n ∣ 0 {displaystyle nmid 0}
  and first ∣ n {displaystyle 1mid n}
  for all n {displaystyle n}
  whole since 0 = 0 ⋅ n {DisplayStyle 0 = 0cdot n}
  and n = n ⋅ first {displaystyle n=ncdot 1}
  .
• 
  first ∣ mcd ⁡ ( a , b ) ∣ a ∣ mcm ⁡ ( a , b ) ∣ 0 {displaystyle 1mid operatorname {mcd} (a,b)mid amid operatorname {mcm} (a,b)mid 0}
  .
• ABCD is divisible between N-1 if and only if A+B+C+D is a multiple of (N-1), provided that ABCD is written at Base N, (n≥ 3). ^{[ 5 ]} ​
• If MCD (a, b) = 1 does not fit ^k = b ^h for any h, k positive whole numbers; Powers of copper are not the same in any case. ^{[ 6 ]} ​
• In any numbering system, to check if n is a multiple of H, divisor of the base, it is enough to analyze the last figure of n. Thus, in the decimal numbering, to know if n is a multiple of 5, it is enough to see if the last figure is 5 or 0. In Base 12, to know if N is divisible between 6, it is enough to see if it ends in 6 or 0. In the hexadecimal system, to check if n is a multiple of 4 it is enough to see that it ends in one of these digits: 0, 4, 8.

Number of divisors [ To edit ]

If the factorization in prime numbers of n It is given by

n = p 1ν1p 2ν2⋯ p kνk{Displaystyle n = P_ {1}^{Nu _ {1}}, P_ {2}^{Nu _ {2}} CDots P_ {k}^{Nu _ {k}}}}

then the number of positive divisors of n is

d ( n ) = ( n 1+ first ) ( n 2+ first ) ⋯ ( n k+ first ) , {Displaystyle D (N) = (Nu _ {1} +1) (Nu _ {2} +1) CDots (Nu _ {k} +1),}

And each of the divisors has the form

p 1μ1p 2μ2⋯ p kμk{displaystyle p_{1}^{mu _{1}},p_{2}^{mu _{2}}cdots p_{k}^{mu _{k}}}

where

0 ≤ m i ≤ n i {Displaystyle 0leq mu _ {i} leq nu _ {i}}

for each

first ≤ i ≤ k . {Displaystyle 1Leq Ileq K.}

^{[ 7 ]} ​

Divisibility criteria [ To edit ]

The following criteria allow you to find out if a number is divisible among another in a simple way, without the need to make the division.

Number	Criterion	Example
first	The number can be divided over itself	5: Because if you divide 5: 1 = 5 and that number is a multiple or divisor of any number.
2	The number ends in a torque figure (0, 2, 4, 6, 8).	378: Because the last figure (8) is even.
3	If the sum of your figures is a multiple of 3.	480: Because 4+8+0 = 12 is a multiple of 3.
4	Its last two digits are 0 or a multiple of 4.	300 and 516 are divisible between 4 because they end in 00 and 16, respectively, the latter being a multiple of 4 (16 = 4*4).
5	The last figure is 0 or 5.	485: Because it ends at 5.
6	It is divisible between 2 and 3.	912: Because it’s torque and 9+1+2 = 12 is a multiple of 3
7	A number is divisible between 7 when, when separating the last figure from the right, multiplying it by 2 and subtracting it from the remaining figures the difference is equal to 0 or is a multiple of 7. Another system: if the sum of the multiplication of the numbers by the series 2,3,1, -2, -3, -1 … gives 0 or a multiple of 7.	34349: We separate 9, and we double it (18), then 3434-18 = 3416. We repeat the process by separating the 6 (341’6) and duplicating it (12), then 341-12 = 329, and again, 32’9, 9*2 = 18, then 32-18 = 14; Therefore, 34349 is divisible between 7 because 14 is a multiple of 7. Example Method 2: 34349: (2*3+3*4+1*3-2*4-3*9) = 6+12+3-8-27 = -14. [ 8 ] ​
8	To know if a number is divisible between 8, it must be verified that its last three figures are divisible between 8. If its last three figures are divisible between 8 then the number is also divisible between 8.
9	A number is divisible by 9 when by adding all its figures the result is a multiple of 9.	504: We add 5+0+4 = 9 and as 9 is a multiple of 9 504 is divisible by 9
ten	The last figure is 0.	4680: Because it ends at 0
11	Adding the figures (of the number) in an odd position on the one hand and those of a position for another. Then the result of both sums obtained is subtracted. If the result is zero or a multiple of 11, the number is divisible between it. If the number has only two figures and these are the same will be a multiple of 11.	42702: 4+7+2 = 13 · 2+0 = 2 · 13-2 = 11 → 42702 is a multiple of 11. 66: Because the two figures are the same. Then 66 is a multiple of 11.
13	A number is divisible between 13 when, when separating the last figure from the right, multiplying it by 9 and subtracting it from the remaining figures the difference is equal to 0 or is a multiple of 13	3822: We separate the last two (382’2) and multiply it by 9, 2*9 = 18, then 382-18 = 364. We repeat the process by separating the 4 (36’4) and multiplying it by 9, 4*9 = 36, then 36-36 = 0; Therefore, 3822 is divisible between 13.
14	A number is divisible between 14 when it is even and divisible between 7	546: We separate the last six (54’6) and bend it, 6*2 = 12, then 54-12 = 42. 42 is a multiple of 7 and 546 is even; Therefore, 546 is divisible between 14.
15	A number is divisible between 15 when it is divisible between 3 and 5	225: It ends in 5 and the sum of its figures is a multiple of 3; Therefore, 225 is divisible between 15.
17	A number is divisible between 17 when, by separating the last figure from the right, multiplying it by 5 and subtracting it from the remaining figures the difference is equal to 0 or is a multiple of 17	2142: Because 214’2, 2*5 = 10, then 214-10 = 204, again, 20’4, 4*5 = 20, then 20-20 = 0; Therefore, 2142 is divisible between 17.
18	A number is divisible between 18 if it is even and divisible between 9 (if it is even and also the sum of its figures is a multiple of 9)	9702: It is even and the sum of its figures: 9+7+0+2 = 18 that is also divisible between 9. And indeed, if we make the division between 18, we will get the rest to be 0 and the quotient 539.
19	A number is divisible between 19 if when separating the figure of the units, multiply it by 2 and add to the remaining figures the result is a multiple of 19.	3401: We separate the 1, bend it (2) and add 340+2 = 342, now we separate the 2, bend it (4) and add 34+4 = 38 which is a multiple of 19, then 3401 is also.
20	A number is divisible between 20 if its last two figures are zeros or multiples of 20. Any torque number that has one or more zeros to the right, is a multiple of 20.	57860: Its last 2 figures are 60 (which is divisible between 20), therefore 57860 is divisible between 20.
23	A number is divisible between 23 if when separating the figure of the units, multiply by 7 and add the remaining figures the result is a multiple of 23.	253: We separate the 3, we multiply it by 7 and add 25+21 = 46, 46 is a multiple of 23 so it is divisible between 23.
25	A number is divisible between 25 if its last two figures are 00, or in a multiple of 25 (25,50.75, …)	650: It is a multiple of 25 for which it is divisible. 400 will also be divisible between 25.
27	A number is divisible by 27, if by dividing it by 3 it gives an exact quotient that is divisible of 9.	11745: Between 3, quotient = 3915; whose figures total 18, then 11745 in divisible between 27.
29	A number is divisible between 29 if when separating the figure of the units, multiply it by 3 and add the remaining figures the result is a multiple of 29.	2262: We separate the last 2, we triple it (6) and add, 226+6 = 232, now we separate the last 2, we triple it (6) and add 23+6 = 29 which is a multiple of 29, then 2262 is also .
thirty first	A number is divisible between 31 if by separating the figure of the units, multiply it by 3 and subtract the remaining figures the result is a multiple of 31.	8618: We separate 8, we triple it (24) and subtract 861-24 = 837, now we separate 7, we triple it (21) and subtract, 83-21 = 62 which is a multiple of 31, then 8618 is also.
50	A number is a multiple of 50 when its last two figures are 00 or 50.	123450: It would be divisible between 50 because it ends at 50.
100	A number will be divisible between 100 if that number ends at 00.	1000: This number will be divisible between one hundred since its last two figures are 00, regardless of the others.
125	A number will be divisible between 125 if its last three figures are 000 or multiple of 125.	3000: It would be divisible between 125 since its last three figures are 000. 4250: This number would also be divisible between 125 since its last three figures are multiple of 125.

Use 1 : There are many versions of the divisibility criteria. For example, for the 13 the criterion is equivalent: when separating the last figure from the right, multiply it by 4 and add it to the remaining figures the sum is equal to 0 or is a multiple of 13.

Use 2 : It is curious that the divisibility criterion by 7 also serves as a divisibility criterion by 3, although obviously the traditional criterion is easier and this is not used: when separating the last figure from the right, multiply it by 2 and subtract it from the figures remaining the difference is equal to 0 or is a multiple of 3.

Use 3 : Although there are similar criteria for any prime number, it is often easier to divide than to apply a complicated criterion (such as 13). However, there is a general criterion that always operates and that in many cases is practical enough: subtract the prime number (or multiples of this) to the figures on the left successively until obtaining zero or that prime number. Thus, the example of 13 could be checked with the following process (we use the 39 = 3*13 to abbreviate steps): 3822 (we subtract 13 twice to the left) → 2522 → 1222 (we subtract 39 three times of the three figures of the left) → 832 → 442 → 52 and when subtracting again 39 we get 52-39 = 13

Use 4: The method does not have to stick only to the process of removing the units. You can take off units and tens. For example: 201 is a multiple of 67. A criterion for 67 would be: we remove the number formed by the tens and units and we subtract it 2 times to the remaining figures, if the result is a multiple of 67, the previous number also also it will be. Example: 66129, we do 661-2 · 29 = 603, now 6 -2 · 3 = 0, then 66129 is a multiple of 67.

A proof of this is as follows: (N-D)/100-2D = (N-D-200D)/100 = (N-201D)/100 = K. If k is a multiple of 67, n will also be since n = 100k+201d.

Use 5: To know if a 3 -digit number is a multiple of 8. The following must 8) or if the number of hundreds is odd and the last two are the result of the difference or sum of a multiple of 8 with 4 (168 → 1 is an odd figure and 68+4 = 72; 72 is a multiple of 8.

Use 6: Any number of three figures, in which the three figures are the same, is a multiple of 3 and 37; In fact it is the multiplication of 37 by the sum of its figures. Example: 333 is a multiple of 37, because 333 = 37*9 (3+3+3 = 9).

Observation [ To edit ]

All the criteria indicated work if the number is written in the decimal numbering system. In another base, this does not always happen. Well 102 ₇ , written on Base 7, ends in torque figure, but it is not divisible between 2. In this case the figures 1+2 = 3 are added; 3 = 1 (Mod 2), Then 102 ₇ is odd (in decimal it is 7 ² +2 = 51).

Other contexts [ To edit ]

Divisibility is possible to deal within the arithmetic properties of

See also [ To edit ]

References [ To edit ]

Bibliography [ To edit ]

• Elemental Arithmetic of Enzo R. Gentile (1985) OAS.
• Theory of Burton W. Jones numbers.
• Fundamentals of Iván VinoGrádov’s numbers theory
• Introduction to Niven and Zuckermann numbers theory
• Arithmetic [i] of L.Gardós (2002), Cultural S.A. Madrid.