Remainder Rules
Lesson · Beginner
Number Theory
Divisibility by 3
The remainder of when divided by 3 is equal to the remainder of the sum of the digits of when divided by 3.
Divisibility by 9
The remainder of when divided by 9 is equal to the remainder of the sum of the digits of when divided by 9.
Divisibility by 11
The remainder of when divided by 11 is equal to the remainder of the alternating sum of its digits.
Example:
From right to left, consider the signs and alternately.
Divisibility by 2, 5, and 10
Only the last digit matters.
Divisibility by 4, 25, and 100
Only the last two digits matter.
Divisibility by 8, 125, and 1000
Only the last three digits matter.
Write
Since
we have
Therefore,
The same argument works modulo 9, because
Hence
and
Since
we have
Therefore,
satisfies
Hence the remainder is determined by the alternating sum of the digits.
Write
So,
Write
So,
Find all digits such that
is divisible by 9.
The digit sum is
We need
Thus
Since is a digit,
Find all digits such that
is divisible by 36.
Since
the divisibility-by-4 condition is automatically satisfied.
For divisibility by 9,
Thus
so
A six-digit number
is divisible by 99. Find all possible pairs .
Since
the number must be divisible by both 9 and 11.
For divisibility by 9, the sum of the digits must be divisible by 9:
Since are digits,
The possible multiples of 9 are 18 and 27. Hence
or
For divisibility by 11, use the alternating sum:
This must be divisible by 11. Since
the only possible multiples of 11 are 0 and . Thus
or
Now combine the possibilities.
If
neither nor gives two digits.
If
and
then adding the equations gives
so
Therefore,
Since
is divisible by both
and
the remainder modulo or depends only on the last digits.
For example, the remainder modulo
depends only on the last four digits.
Look for simple patterns among
If
we can split the number into -digit blocks and add the blocks.
If
we can use an alternating sum of -digit blocks.
When using the rule for 11, the alternating sum may be negative.
For example, if
then
Therefore, the remainder is