In this section, we will explore some rules of divisibility for some positive integers. These rules can be easily extended to all the integers by dropping the sign.
Let .
Then
which implies
Thus we can express as , where is the ones digit of , and .
Divisibilty by :
Let . iff . In other words, divides an integer iff the ones digit of the integer is either or .
Proof:
Since , , and by divisibility theorem I, iff .
Divisibility by
iff . In other words, divides an integer iff the ones digit of the integer is either or .
Proof:
Since , , and by divisibility theorem I, iff .
Divisibility by
iff . In other words, divides an integer iff the ones digit of the integer is .
Proof:
Since , , and by divisibility theorem I, iff .
Divisibility by
iff .
Proof:
Let be an integer. Then
, which implies
.
Since , , and by divisibility theorem I, iff .
Divisibility by of a two digit number
iff
Divisibility by
iff .
Proof:
Let be an integer. Then
, which implies
.
Since , , and by divisibility theorem I, iff .
A similar argument can be made for divisibility by , for any positive integer .
Example :
Using divisibility tests, check if the number is divisible by:
Solution:
is not divisible by
Rule: The one's digit of the number has to be either a or a
Since the last digit is not or is not divisible by
2. is divisibleby
Rule: The last two digits of the number have to be divisible by 4.
The last two digits of is
Since is divisible by is divisible by also.
3. is not divisible by
Rule: The last three digits of the number have to be divisible by 8.
The last three digits of is Since is not divisible by the original number, is not divisible by either.
Notice the following pattern
Divisibility by
iff divides sum of its digits.
Proof:
Let .
Then
which implies
Example :
Find the possible values for the missing digit , if is divisible by
Solution:
Consider the following:
The divisibility rule for the number 3 is as follows: If the sum of the digits in the whole number is a number divisible by 3, then the larger, original number is also.
The sum of the digits is Since , or . Hence or .
Divisibility by
iff divides sum of its digits.
Divisibility by
iff divides the absolute difference between , where , where is the ones digit of , and .
Proof:
Assume Which implies
Consider .
Thus if and only if
Divisibility by
iff divides the absolute difference between the alternate sum.