You might think that if you divide two numbers and get an integer result, then the first number is divisible by the second. But what about negative numbers? Or zero? And why does 5 go into 10 twice as easily as 4 goes into 8?
Before anything else: divisibility is not just about division. It’s actually a property that some numbers have in relation to others. Specifically, if m divides n evenly (i.e., without leaving any remainder), then we say that m is a factor or divisor of n, and n is a multiple of m.
For example, 3 goes into 6 exactly twice: once with a remainder of zero, and again with a remainder of zero. So 3 is a factor (and therefore a divisor) of 6, and 6 is a multiple of 3. Easy peasy!
But what about negative numbers? Can they be factors or multiples too? Of course they can! Let’s say we have -12 and we want to know if it’s divisible by 4. Well, if we divide -12 by 4 (using long division), we get:
-3 | 12
—
4
______
8
So the remainder is zero, which means that 4 goes into -12 exactly three times. Therefore, 4 is a factor of -12 (and also a divisor), and -12 is a multiple of 4.
But what about when we try to divide two negative numbers? For example, let’s say we want to know if -8 is divisible by -3. If we do the same long division as before:
– | 8
—
3
______
2
We get a remainder of 2! This means that -3 does not go into -8 exactly, so it’s not a factor or divisor. However, we can still say that -8 is divisible by the product of two negative numbers (-3 and -2), since:
– | 8
—
3
______
2
If we multiply both sides by -1 (which flips the sign on everything):
1 | -8
——
-3/1 | 3
—–
2
We get:
-6 | 0
—
3
______
6
So we can see that (-3) x (-2) is a factor of -8, and therefore -8 is divisible by the product of two negative numbers.
But what about zero? Can it be a divisor or factor too? Well, technically speaking, zero goes into any number exactly 0 times (since there’s no remainder). So we can say that zero is a factor of every number, but not in the traditional sense. It doesn’t really “divide” anything; instead, it just makes everything divisible by itself!
So now you know: divisibility isn’t always as straightforward as division.