# divides

We say one integer divides another if it does so evenly, that is with a remainder of zero (we sometimes say, "with no remainder," but that is not technically correct). More formally, mathematicians write:

Ifaandbare integers (withanot zero), we sayadividesbif there is an integercsuch thatb=ac.

We use this concept enough that it has its own symbols:

a|bmeans adividesb.abmeans adoes not divideb.

The integers that divide *a* are called the divisors of *a*.

You might want try your hand at proving the following basic properties which hold for all integers *a*, *b*.
*c* and *d*:

*a*| 0, 1 |*a*,*a*|*a*.*a*| 1 if and only if*a*=+/-1.- If
*a*|*b*and*c*|*d*, then*ac*|*bd*. - If
*a*|*b*and*b*|*c*, then*a*|*c*. *a*|*b*and*b*|*a*if and only if*a*=+/-*b*.- If
*a*|*b*and*b*is not zero, then |*a*|__<__|*b*|. - If
*a*|*b*and*a*|*c*, then*a*| (*bx*+*cy*) for all integers*x*and*y*.

Finally, suppose *p* is a prime and *k* is
a positive integer. A notation that is quickly
gaining acceptance is to write *p ^{k}* ||

*a*to indicate that

*p*divides

^{k}*a*, but

*p*

^{k+1}does not.

**See Also:** GCD, Prime, RelativelyPrime

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