Divisibility rules for the numbers 2 through 10 (Is a number divisible by 2? By 3? By 4? etc.)

What does it mean for a number to be “divisible” by another number?

When we talk about the “divisibility” of a whole number, we’re just talking about the whole numbers that divide evenly into it. As an example, ???5??? divides evenly into ???15??? three times, since ???15\div5=3???, which means we can say that ???15??? is “divisible” by ???5???.

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If we want to be technical, ???15??? is divisible by ???5??? because when we do the division ???15\div5???, the answer we get (???3???) is a whole number. That’s the technical definition of divisibility: The result of the division must be a whole number.

Another way to say this is that we must get a remainder of ???0???. Since we often think of ???0??? as “nothing,” we sometimes say that there’s no remainder when the remainder is ???0???. So a third way to say that one whole number is divisible by another is that we don’t get a remainder when we do the division.

Here’s a counterexample. Is ???9??? divisible by ???4???? If we divide ???9??? by ???4???, we know ???4??? goes into ???9??? two times, and that gets us up to ???8???, but then we have a remainder of ???1???. In other words, because we have a remainder (other than ???0???), our answer isn’t a whole number. We get a whole number as the answer for ???8\div4??? (the answer is the whole number ???2???), and we get a whole number as the answer for ???12\div4??? (the answer is the whole number ???3???), but we don’t get a whole number as the answer for ???9\div4???. Therefore, we can say that ???9??? is not divisible by ???4???. But as we just saw, ???8??? and ???12??? are both divisible by ???4???.

The following list of divisibility rules are a shorthand way of determining if a particular integer is divisible by another, without actually performing the division.

Divisible by ???2??? if the last digit is ???0,\ 2,\ 4,\ 6,\ 8???

Divisible by ???3??? if the sum of the digits is divisible by ???3???

Divisible by ???4??? if the last two digits are divisible by ???4???

Divisible by ???5??? if the last digit is ???0,\ 5???

Divisible by ???6??? if divisible by ???2??? and ???3???

Divisible by ???7??? if ???5\times??? last digit + rest of the number is divisible by ???7???

Divisible by ???8??? if the last three digits are divisible by ???8???

Divisible by ???9??? if the sum of the digits is divisible by ???9???

Divisible by ???10??? if the last digit is ???0???