The divisibility of a positive number by an integer divisor can be determined by
some divisibility rule. The divisibility rules are usually a test to test the digits of a number without
performing the division directly.
Divisibility Test
A base 10 number, a, can be expressed in terms of digits ai, that is
Some
standard divisibility tests are as following
Testing of Ending Digit Block
A base 10 number can also be expressed as the sum of two parts. If the number
a is divided by a divisor, d, the divisor, d, must divides both parts. In other
words, for a divisor dividing the power of ten, the divisor divides the given
number if and only if the divisor divides the ending digits.
For example,
Therefore, this is the divisibility test for numbers by number with prime factor
2 and 5.
Testing of
the Sum of Digit Blocks
Instead of expressing a base 10 number as the sum of two parts, a base 10 number
can also be expressed in forms of parts of fixed size digit blocks. By deducting
the sum of digit blocks from the given number, the coefficients of remaining
digit blocks become a repeated pattern of digit 9. If the number a is divided by
a divisor, d, the divisor, d, must divides the sum of digit blocks and all
parts of fixed size digit blocks. In other words, for a divisor dividing the
power of ten minus 1, the divisor divides the given number if and only if the
divisor divides the sum of digit blocks.
For example,
Therefore, this is the divisibility test for numbers by number with prime factor
of a digit block with dight 9s, i.e. 9, 99, 999,....
Testing of
the Alternating Sum of Digit Blocks
Similar to testing of
the sum of digit blocks, a base 10 number can be expressed in forms of parts of fixed size digit blocks.
Instead of refering to the digit block of "9...9", a digit block of the form
"10...01". The relation can be determined by modular arithmetic. By using
modular arithmetic, the coefficients of
digit blocks become an alternating sum of digit blocks. If a number a is divided by
the divisor, d, the divisor, d, must divide the alternating sum of digit blocks
also. In other words, for a divisor dividing the
power of ten plus 1, the divisor divides the given number if and only if the
divisor divides the alternating sum of digit blocks.
The same relation can also be determined algebrically as in testing of the sum
of digit blocks. The coefficient of all negative terms are always divided by the
first coefficient 10i+1, i.e. 10i+1|10i+2x+1. Besides, The coefficient of all positive terms
are also always divided by the first coefficient 10i+1,
i.e. 10i+1|10i+2x+1-1.
For example,
Eliminate from the right digit
block
Instead of testing the ending digit block, the ending digit block can also be
eliminated by reducing the coefficient of the front digit block from the
powers of ten to 1 so as to reduce the given number by a digit block from the
right.
The same relation can also be determined
using modular arithmetic. The integer multipler u is expressed as an inverse of
10i modulo d.
For example,
Eliminate from the
left digit block
Similar
to eliminating from the right dight block, the beginning digit block can also be
eliminated by reducing the coefficient of the front digit block by
powers of ten so as to reduce the given number by a digit block from the left.
The integer multipler u can also be determined algebrically
For example,
Factor the Divisor
For a composite divisor, the divisibility tests for each prime factor of the
composite divisor can be used to test the given numebr one by one
seperately.
Divisibility Test for a Numberr a Number
The divisibility of an integer number for some integers can be examined by some
rules:
2 - number with the last digit is even
3 - number with the sum of digits is divisible by 3
4 - number with the number of the last two digits is divisible by 4
5 - number with the last digit is 5 or 0
6 - number is divisible by both 2 and 3
7 - number with the number without the last digit minus the double of the last
digit is equal to 0 or divisible by 7
8 - number with the number of the last three digits is divisible by 8
9 - number with the sum of digits is divisible by 9
10 - number with the last digit is 0
11 - number with the sum of every second digit minus the sum of other digits is
equal to 0 or divisible by 11
12 - number is divisible by both 3 and 4
25 - number with the number of the last two digit is 00, 25, 50, or 75