Divisibility Rules

Dividing by 3

Add up the digits: if the sum is divisible by three, then the number is as well. Examples:
1. 111111: the digits add to 6 so the whole number is divisible by three.
2. 87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.
Why does the 'divisibility by 3' rule work?
Look at a 2 digit number: 10a+b=9a+(a+b). We know that 9a is divisible by 3, so 10a+b will be divisible by 3 if and only if a+b is. Similarly, 100a+10b+c=99a+9b+(a+b+c), and 99a+9b is divisible by 3, so the total will be if a+b+c is.
This explanation also works to prove the divisibility by 9 test. It is originates from modular arithmetic ideas.

  Dividing by 4

Look at the last two digits. If the number formed by its last two digits is divisible by 4, the original number is as well.
Examples:
1. 100 is divisible by 4.
2. 1732782989264864826421834612 is divisible by four also, because 12 is divisible by four.

  Dividing by 5

If the last digit is a five or a zero, then the number is divisible by 5.

  Dividing by 6

Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well.

Another easy way to tell if a [multi-digit] number is divisible by six . . . is to look at its [ones digit]: if it is even, and the sum of the [digits] is a multiple of 3, then the number is divisible by 6.

  Dividing by 7

To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number.
Example: If you had 203, you would double the last digit to get six, and subtract that from 20 to get 14. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again.

Another methods is:

To know if a number is a multiple of seven or not, we can use also 
3 coefficients (1 , 2 , 3). We multiply the first number starting 
from the ones place by 1, then the second from the right by 3, 
the third by 2, the fourth by -1, the fifth by -3, the sixth by -2, 
and the seventh by 1, and so forth.

Example: 348967129356876. 

6 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16 
means the number is not multiple of seven.

If the number was 348967129356874, then the number is a multiple 

 of seven because instead of 16, we would find 14 as a result, which is a
  multiple of 7. So the pattern is as follows: for a number onmlkjihgfedcba,
  calculate a + 3b + 2c - d - 3e - 2f + g + 3h +



2i - j - 3k - 2l + m + 3n + 2o.

Example:  348967129356874.

Below each digit let me write its respective figure.

3  4  8  9  6  7  1  2  9  3  5  6  8  7  6 
2  3  1 -2 -3 -1  2  3  1 -2 -3 -1  2  3  1

(3×2) + (4×3) + (8×1) + (9×-2) + (6×-3) + (7×-1) + 
(1×2) + (2×3) + (9×1) + (3×-2) + (5×-3) + (6×-1) + 
(8×2) + (7×3) + (6×1) =  16 -- not a multiple of seven.

  Dividing by 8

Check the last three digits. Since 1000 is divisible by 8, if the last three digits of a number are divisible by 8, then so is the whole number.
Example: 33333888 is divisible by 8; 33333886 isn't. How can you tell whether the last three digits are divisible by 8?
If the first digit is even, the number is divisible by 8 if the last two digits are. If the first digit is odd, subtract 4 from the last two digits; the number will be divisible by 8 if the resulting last two digits are. So, to continue the last example, 33333888 is divisible by 8 because the digit in the hundreds place is an even number, and the last two digits are 88, which is divisible by 8. 33333886 is not divisible by 8 because the digit in the hundreds place is an even number, but the last two digits are 86, which is not divisible by 8.

  Dividing by 9

Add the digits. If that sum is divisible by nine, then the original number is as well. This holds for any power of three.

  Dividing by 10

If the number ends in 0, it is divisible by 10.

  Dividing by 11

Take any number, such as 365167484. Add the first, third, fifth, seventh,.., 

digits.....3 + 5 + 6 + 4 + 4 = 22
Add the second, fourth, sixth, eighth,.., digits.....6 + 1 + 7 + 8 = 22
If the difference, including 0, is divisible by 11, then so is the number.
22 - 22 = 0 so 365167484 is evenly divisible by 11.

  Dividing by 12

Check for divisibility by 3 and 4.

Read More

Number Notations for Maths

Hierarchy of Decimal Numbers

Number	Name	How many
0	zero
1	one
2	two
3	three
4	four
5	five
6	six
7	seven
8	eight
9	nine
10	ten
20	twenty	two tens
30	thirty	three tens
40	forty	four tens
50	fifty	five tens
60	sixty	six tens
70	seventy	seven tens
80	eighty	eight tens
90	ninety	nine tens

Number	Name	How Many
100	one hundred	ten tens
1,000	one thousand	ten hundreds
10,000	ten thousand	ten thousands
100,000	one hundred thousand	one hundred thousands
1,000,000	one million	one thousand thousands

Some people use a comma to mark every 3 digits. It just keeps track of the digits and makes the numbers easier to read.
Beyond a million, the names of the numbers differ depending where you live. The places are grouped by thousands in America and France, by the millions in Great Britain and Germany.

Name	American-French	English-German
million	1,000,000	1,000,000
billion	1,000,000,000 (a thousand millions)	1,000,000,000,000 (a million millions)
trillion	1 with 12 zeros	1 with 18 zeros
quadrillion	1 with 15 zeros	1 with 24 zeros
quintillion	1 with 18 zeros	1 with 30 zeros
sextillion	1 with 21 zeros	1 with 36 zeros
septillion	1 with 24 zeros	1 with 42 zeros
octillion	1 with 27 zeros	1 with 48 zeros
googol	1 with 100 zeros
googolplex	1 with a google of zeros

Fractions
Digits to the right of the decimal point represent the fractional part of the decimal number. Each place value has a value that is one tenth the value to the immediate left of it.

Number	Name	Fraction
.1	tenth	1/10
.01	hundredth	1/100
.001	thousandth	1/1000
.0001	ten thousandth	1/10000
.00001	hundred thousandth	1/100000

Examples:
0.234 = 234/1000 (said - point 2 3 4, or 234 thousandths, or two hundred thirty four thousandths)
4.83 = 4 83/100 (said - 4 point 8 3, or 4 and 83 hundredths)
SI Prefixes

Number Prefix Symbol 10 1 deka- da 10 2 hecto- h 10 3 kilo- k 10 6 mega- M 10 9 giga- G 10 12 tera- T 10 15 peta- P 10 18 exa- E 10 21 zeta- Z 10 24 yotta- Y

Number Prefix Symbol 10 -1 deci- d 10 -2 centi- c 10 -3 milli- m 10 -6 micro- u (greek mu) 10 -9 nano- n 10 -12 pico- p 10 -15 femto- f 10 -18 atto- a 10 -21 zepto- z 10 -24 yocto- y

Roman Numerals

I=1		(I with a bar is not used)
V=5		_ V=5,000
X=10		_ X=10,000
L=50		_ L=50,000
C=100		_ C = 100 000
D=500		_ D=500,000
M=1,000		_ M=1,000,000

Examples:

1 = I 2 = II 3 = III 4 = IV 5 = V 6 = VI 7 = VII 8 = VIII 9 = IX 10 = X

11 = XI 12 = XII 13 = XIII 14 = XIV 15 = XV 16 = XVI 17 = XVII 18 = XVIII 19 = XIX 20 = XX 21 = XXI

25 = XXV 30 = XXX 40 = XL 49 = XLIX 50 = L 51 = LI 60 = LX 70 = LXX 80 = LXXX 90 = XC 99 = XCIX

There is no zero in the roman numeral system.
The numbers are built starting from the largest number on the left, and adding smaller numbers to the right. All the numerals are then added together.
The exception is the subtracted numerals, if a numeral is before a larger numeral, you subtract the first numeral from the second. That is, IX is 10 - 1= 9.
This only works for one small numeral before one larger numeral - for example, IIX is not 8, it is not a recognized roman numeral.
There is no place value in this system - the number III is 3, not 111.
Number Base Systems

Decimal(10)	Binary(2)	Ternary(3)	Octal(8)	Hexadecimal(16)
0	0	0	0	0
1	1	1	1	1
2	10	2	2	2
3	11	10	3	3
4	100	11	4	4
5	101	12	5	5
6	110	20	6	6
7	111	21	7	7
8	1000	22	10	8
9	1001	100	11	9
10	1010	101	12	A
11	1011	102	13	B
12	1100	110	14	C
13	1101	111	15	D
14	1110	112	16	E
15	1111	120	17	F
16	10000	121	20	10
17	10001	122	21	11
18	10010	200	22	12
19	10011	201	23	13
20	10100	202	24	14

Each digit can only count up to the value of one less than the base. In hexadecimal, the letters A - F are used to represent the digits 10 - 15, so they would only use one character.

Read More