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Number theoryIn mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we usually write 7 | 42. Divisors can be positive or negative. The positive divisors of 42 are {1, 2, 3, 6, 7, 14, 21, 42}.
Some special cases: 1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd.
The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient.
1 Rules for small divisors
There are some rules which allow to recognize small divisors of a number from the number's decimal digits:
A divisibility rule is a rule you can use to determine a number's divisibility by another number. In decimal, the divisibility rules are:
- a number is divisible by 2 iff the last digit is divisible by 2
- a number is divisible by 3 iff the sum of its digits is divisible by 3
- a number is divisible by 4 iff the number given by the last two digits is divisible by 4
- a number is divisible by 5 iff the last digit is 0 or 5
- a number is divisible by 66 six is the natural number following 5 and preceding 7. Prefixes for 6 include hexa- ( Greek) and sex- ( Latin). The SI prefix for 10006 is exa (E), and for its reciprocal atto (a). Evolution of the glyph The evolution of our modern glyph for 6 appears r iff it is divisible by 2 and by 3
- a number is divisible by 77 seven is the natural number following 6 and preceding 8. Seven is the smallest positive integer requiring two syllables in English. Prefixes for 7 include hepta- ( Greek) and septua- Latin. The SI prefix for 10007 is zetta (Z), and for its reciprocal ze iff the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2×4 = 28 is divisible by 7)
- a number is divisible by 88 eight is the natural number following 7 and preceding 9. Prefixes for 8 include octa- ( Greek) and octo- ( Latin. The SI prefix for 10008 is yotta (Y), and for its reciprocal yocto (y). Evolution of the glyph In the beginning, various groups in India wr iff the number given by the last three digits is divisible by 8
- a number is divisible by 99 nine is the natural number following 8 and preceding 10. Prefixes for 9 include ennea- ( Greek) and nona- ( Latin. Evolution of the glyph In the beginning, various Indians wrote 9 in a way that pretty much looks like our modern closing question mark, bu iff the sum of its digits is divisible by 9
- a number is divisible by 1010 ten is the natural number following 9 and preceding 11. Prefixes for 10 include deca- ( Greek) and deci- ( Latin). In mathematics Ten is a composite number, its proper divisors being 1, 2 and 5. Ten is the base of the decimal numeral system, by far the iff the last digit is 0
- a number is divisible by 1111 eleven is the natural number following 10 and preceding 12. Eleven is the smallest positive integer requiring three syllables in English. In mathematics Eleven is the 5th smallest prime number, and also a repunit prime. Although it is necessary for n t iff the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)
- a number is divisible by 1212 twelve is the natural number following 11 and preceding 13. In mathematics Twelve is a composite number, its proper divisors being 1, 2, 3, 4 and 6. Twelve is also a highly composite number, the next one being 24. Twelve being the product of three and iff it's divisible by 3 and by 4
- A number is divisible by 13
- iff the result of adding 4 times the last digit to the original number.
- iff the result of subtracting 9 times the last digit from the number with the last digit removed is divisible by 13 (e.g. 858 is divisible by 13 since 85-9×8 = 13 is divisible by 13).
- A number is divisible by 14 iff it's divisible by 2 and by 7.
- A number is divisible by 15 iff it's divisible by 3 and by 5.
