The Difference Between Divides and Divisible
This terminology is important, because we use it in the context of factoring and dividing polynomials. Let’s talk first about it in the context of whole numbers and then we’ll see how we might use it in algebraic situations.
When we divide one whole number by another whole number, either the remainder is 0 or it is some positive whole number.
If the remainder when dividing y by x is 0, then we say:
a) x divides y
b) y is divisible by x
Samples
a) 7 divides 42 because 42 divided by 7 = 6 with remainder 0
b) 12 is divisible by 3, because 12 divided by 3 = 4 with remainder 0
c) 2 does not divide 7 because 7 divided by 2 = 3 with remainder 1
d) 13 is not divisible by 5 because 13 divided by 5 = 2 with remainder 3
Note: y divided by x means the same as y ÷ x or 
If x divides y, then we say:
1. x is a factor of y
2. x is a divisor of y
Samples
a) 12 is a factor of 36 since 12 divides 36
b) 7 is a divisor of 56 since 7 divides 56
c) 8 is not a factor of 18 since 8 does not divide 18
d) 9 is not a divisor of 24 since 9 does not divide 24
If x divides y, then we say:
a) y is a multiple of x
b) y is divisible by x
Samples
a) 63 is a multiple of 7, since 7 divides 63
b) 32 is divisible by 8 since 8 divides 32
c) 45 is not a multiple of 6 since 6 does not divide 45
d) 37 is not divisible 9 since 9 does not divide 37
Examples
a) Write two true statements using “divides” and the numbers 3, 7, and 21
b) Write two true statements using “divisible” and the numbers 4, 9, and 36.
Solution
a) 3 divides 21
7 divides 21
b) 36 is divisible by 4
36 is divisible by 9
In more general algebra situations, we might say, for example, that
a) x is a factor of x2 + xy since x(x + y) = x2 + xy
b) x divides x2 + xy
c) x2 + xy is divisible by x