27 Divisibility Tests

Definition and Notation for Divisibility

Later in this module, we will explore prime numbers, composite numbers, the greatest common factor of two or more numbers, and the least common multiple of two or more numbers. Factoring is the method used to find the prime factorization of a composite number, and it can also be used to find the greatest common factor and least common multiple of a set of numbers. One of the problems in factoring large numbers is that sometimes it isn’t clear if it is prime or composite. In other words, it isn’t clear whether it has any factors other than [latex]1[/latex] or itself. Most of us know that if the last digit of a numeral is even, then [latex]2[/latex] will divide into it; or if it ends in [latex]0[/latex], [latex]10[/latex] will divide into it; or if it ends in [latex]0[/latex] or [latex]5[/latex], that [latex]5[/latex] will divide into it. Other than that, people may even have trouble determining if relatively small numbers are prime. For instance, many people think [latex]91[/latex] is prime, but in fact it is not ([latex]7[/latex] is one of its factors). Knowing some divisibility tests makes the task easier, so we’ll soon take some time to discuss divisibility tests for several numbers. First, we need to go over the concept and notation concerning divisibility.

Okay, so what exactly does “there exists a whole number, n, such that [latex]a \times n[/latex] [latex]= b[/latex]” mean? Well, it means that the first number times some whole number equals the second number. Or you can think “[latex]a[/latex] divides [latex]b[/latex]” is true if [latex]b \div a[/latex] is a whole number.

That is, [latex]a|b[/latex] tells us the divisibility of whole number [latex]b[/latex] by its divisor [latex]a[/latex]. According the definition, if [latex]b[/latex] is divisible by [latex]a[/latex], or [latex]a[/latex] divides [latex]b[/latex], then the divisor [latex]a[/latex] of it must also be one of its factors. In other words, here the term divisor is only used for the number that can divide the dividend evenly. As a result, in number theory, people may use “factor” and “divisor” interchangeably, considering greatest common divisor and greatest common factor as an example. However, we should be very cautious that, in a context of division as an operation of whole numbers, divisor of a division problem can be any non-zero number that we use to divide a given dividend.

For example, in the division problem [latex]22 \div 5[/latex], which equals [latex]4[/latex] with remainder [latex]2[/latex], we say [latex]5[/latex] is a divisor of the division. But can we say “[latex]5[/latex] is a divisor of [latex]22[/latex]“? No, because [latex]5[/latex] is not a factor of [latex]22[/latex]. In other words, we can say, [latex]22[/latex] is not divisible by [latex]5[/latex]. Intuitively, this means [latex]22[/latex] cannot be divided by [latex]5[/latex] evenly, because there is a remainder [latex]2[/latex]. We describe this fact as “[latex]5[/latex] is not a divisor of [latex]22[/latex]“, written as [latex]5\not{|} 22[/latex].

More discussion on the notations for “divides” and “divided by”:

When you see [latex]12/3[/latex], this means “[latex]12[/latex] divided by [latex]3[/latex]“. The slash that slants to the right is another way to write the division sign, [latex]\div[/latex]. So, [latex]12/3[/latex] means the same as [latex]12 \div 3[/latex], read as “[latex]12[/latex] divided by [latex]3[/latex]“. It is a solvable division problem with [latex]4[/latex] as its answer: [latex]12 \div 3[/latex] = [latex]4[/latex]. We read this equation as a statement “[latex]12[/latex] divided by [latex]3[/latex] equals 4.” From a linguistic perspective, “[latex]12 \div 3[/latex]“, as a number described by a prepositional phrase, serves as the subject of the statement, “[latex]=[/latex]” as the verb, and the answer [latex]4[/latex] is the object.

Differently, if I say “[latex]3[/latex] divides [latex]12[/latex]“, I am making a stand alone statement, “[latex]3[/latex] divides [latex]12[/latex],” in which the subject is [latex]3[/latex], “divides” is the verb, and [latex]12[/latex] is the object. It is not a division problem that needs to be solved. It is a statement that happens to be true. The symbol used to represent the verb “divides” is a vertical line. So, “[latex]3[/latex] divides [latex]12[/latex]” can be written “[latex]3|12[/latex]“. Again, this is a statement, a fact, not a division problem. The way to express “does not divide” is to put a slash through the symbol: [latex]\not{|}[/latex].

How to decide if a number divides another number?

Let’s get back to why “[latex]3[/latex] divides [latex]12[/latex]” is a true statement. According to the definition, you can be formal and ask yourself:

Is there a whole number [latex]n[/latex] such that [latex]3 \times n[/latex] = [latex]12[/latex]?

Or, you can simply ask yourself:

Is [latex]3[/latex] a factor (or divisor) of [latex]12[/latex]?

In either case, the answer is yes, so the statement is true. In Example 1 below we will practice more the terms of “divides” and “divisible”.

Let’s work a little more on the difference between a statement using “divides”, and an actual division problem, especially when the math notations are used. The example below helps us practice more using the symbol [latex]|[/latex]for “divides” and [latex]/[/latex] for “divided by“. For each item in the example, you can read the given numbers and symbol in between, then decide its meaning.

Below we will discuss properties of divisibility and do some proofs for them. Before doing so, let’s go back to the formal definition of “divides”:

[latex]a|b[/latex]if there exists a whole number, [latex]n[/latex], such that [latex]an = b[/latex].

We need to use this formal notation in order to do some proofs.

For all the our choices, the statement, “If a|b and a|c, then a|(b+c)”, happens to be true. Now to argue that it is always true, we must write a proof to prove it. Below is one way to write a formal proof.

Start by picking any number for a, like a = 3 and picking a number for the sum b + c, for which 3 is a factor, like 15 or 18, etc. Let b + c = 15. Note that there are many combinations of numbers that add up to 15: 1 + 14, 2 + 13, 3 + 12, 7 + 8, etc. You are being asked if 3 divides into the sum of any two whole numbers, will it necessarily divide into each individual addend as well? For instance, if you broke 15 up as the sum of 9 and 6, this would be the statement: “If 3|(9 + 6), then 3|9 and 3|6.” Using these numbers, it is true and you haven’t found a counterexample. Try splitting up 15 another way, perhaps as the sum of 10 and 5. Then, the statement becomes: “If 3|(10 + 5), then 3|10 and 3|5.” The answer is no, and therefore these numbers may be used as a counterexample, which proves the statement is false. Remember: In order for a statement to be true, it must be for all values of a and b. On the other hand, only one counterexample showing it is not true is sufficient to prove a statement is false. As a conclusion, the statement “If a|(b+c), then a|b and a|c.” is false.

Similar to the sum property of divisibility, we can conclude the product property of divisibility as below.

All counting numbers can be considered factors of zero. In other words, for all counting numbers, m, m|0 is always true since there is always some number times m that equals zero, namely zero itself.

Assume n is a positive whole number. The following are divisibility tests to determine what numbers from 2 to 12 divide into n, or what numbers are factors of n. We divide the numbers from 2 to 12 into several families according to their divisibility tests.

Family of 10

The divisibility by 10 is the most obvious one: if the last digit of a number is 0, then the number is divisible by 10, that is, 10 divides this number. The family of 10 includes 10 and its two factors 2 and 5. Similar to the divisibility by 10, we can observe the last digit of a number to determine if the number is divisible by 2 or 5. For example, if the last digit of a number is an even number, then the number is divisible by 2. Similarly, if the last digit of a number is 0 or 5, then we know it is divisible by 5.

Although 2 as a factor of 10 is listed in the family of 10, it leads a family itself. The divisibility for 2 can be re-stated as

2|n if 2|(the last digit of n).

Similarly, the divisibility of a number by 4 can be determined by only observing its last two digits, and the divisibility of a number by 8 can be determined by only observing its last three digits.

The divisibility tests for members in this family are about the digital roots we discussed in the last section. Recall that the digital root of a number is the remainder obtained when a number is divided by 9, and that the digital root of a number can be calculated by adding its digits and casting out 9s. Therefore, “the digital root of a number is 0” means that 9 divides this number. Similarly, the divisibility of a number by 3 can be determined by the divisibility of its digital root by 3.

Example 8 helps us to practice the divisibility test for 3.

Both 6 and 12 have multiple factors other than 1 and themselves. We can call this family “multi-factor family”. For 6, we can use divisibility tests for 2 and 3 to determine if a number is divisible by 6. For 12 as product of 3 and 4, we use the tests for 3 and 4.

Important Note: A number is divisible by 6 only if it passes the divisibility test for both 2 and 3. If it doesn’t pass one of the tests, then it is not divisible by 6. To show it is divisible by 6, we must show it passes both of the tests. Similarly, to show a number is divisible by 12, we must show it passes both of the tests for 3 and 4. On the contrary, if it doesn’t pass one of the tests for 3 or 4, we can conclude that it is not divisible by 12.

Family of 7 and 11

7 and 11 do not share a similar test; they are put into the same family because of the uniqueness of the their divisibility tests.

This test is more confusing to describe than to do. What you do is add up every other digit. Then, add up the ones you skipped. Then, subtract these two numbers and see if 11 divides this number.

Listed below are the divisibility tests that you should know. It’s important to realize that each of the tests only work for the number specified. In other words, you can’t use the divisibility test for 3 to determine is 7 divides a number. The divisibility test for 7 has nothing to do with digital roots!

Further exploration: Can you develop divisibility rules for 14, 15, and 18?