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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Divisibility rule | 1/8 | https://en.wikipedia.org/wiki/Divisibility_rule | reference | science, encyclopedia | 2026-05-05T08:13:56.367542+00:00 | kb-cron |
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.
== Divisibility rules for numbers 1−30 == The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means. For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.
== Step-by-step examples ==
=== Divisibility by 2 === First, take any number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2. Example
376 (The original number) 37 6 (Take the last digit) 6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2) 376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2)
=== Divisibility by 3 or 9 === First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9). Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the remainder of the original number if it were divided by nine (unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero). This can be generalized to any standard positional system, in which the divisor in question then becomes one less than the radix; thus, in base-twelve, the digits will add up to the remainder of the original number if divided by eleven, and numbers are divisible by eleven only if the digit sum is divisible by eleven. Example.
492 (The original number) 4 + 9 + 2 = 15 (Add each individual digit together) 15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large: 1 + 5 = 6 (Add each individual digit together) 6 ÷ 3 = 2 (Check to see if the number received is divisible by 3) 492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)
=== Divisibility by 4 === The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two-digit number that is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits. Alternatively, one can just add half of the last digit to the penultimate digit (or the remaining number). If that number is an even natural number, the original number is divisible by 4. Also, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4. Example. General rule
2092 (The original number) 20 92 (Take the last two digits of the number, discarding any other digits) 92 ÷ 4 = 23 (Check to see if the number is divisible by 4) 2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4) Second method
6174 (the original number) check that last digit is even, otherwise 6174 can't be divisible by 4. 61 7 4 (Separate the last 2 digits from the rest of the number) 4 ÷ 2 = 2 (last digit divided by 2) 7 + 2 = 9 (Add half of last digit to the penultimate digit) Since 9 is not even, 6174 is not divisible by 4 Third method
1720 (The original number) 1720 ÷ 2 = 860 (Divide the original number by 2) 860 ÷ 2 = 430 (Check to see if the result is divisible by 2) 1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)