They have a converse relationship. A factor is a number that divides into another, with no remainders. A multiple is something that can be multiplied by another number, to reach the original number that you had.
Thus: Original number / a factor = a multiple
Multiple * (the correct factor) = original number.
"the correct factor", because most number have more than one factor. But only one factor * multiple will give the original number.
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They both describe relationships between numbers.
Factors go into numbers, numbers go into multiples.
15 is both a multiple and a factor of itself.
It can be both. 6 is a factor of 12 and a multiple of 3.
A factor is what goes into a number, a multiple is what a number goes into. 3 is a factor of 9 27 is a multiple of 9.