# Divisibility property in mathematics

One of the most often misunderstood concepts in math is divisibility. Divisibility is a different operation than division and the two are often confused.

The solution describes the seven key properties of divisibility and proves them mathematically to show the reader that they are true and why they are important.

The seven divisibility properties are as follows:

(a) a|0, a|a

(b) a|1 if and only if a = ±1

(c) if a|b and c|d, then ac|bd (for c?0)

(d) if a|b and b|c, then a|c (for b?0)

(e) (a|b and b|a) if and only if a = ±b

(f) if a|b and b?0, then |a|?|b|

(g) if a|b and a|c, then a|(bx+cy) for any integers x,y.

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#### Solution Preview

Prove all the parts of the divisibility theorem, giving the basic properties of divisibility. Namely, show that for any integers a,b,c,d, with a≠0:

(a) a|0, a|a

(b) a|1 if and only if a=±1

(c) if a|b and c|d, then ac|bd (for c≠0)

(d) if a|b and b|c, then a|c (for b≠0)

(e) (a|b and b|a) if and only if a=±b

(f) if a|b and b≠0, then |a|≤|b|

(g) if a|b and a|c, ...

#### Solution Summary

This solution explains divisibility property in number theory. It also explains the differences between division and divisibility. This solution also provides proofs for properties of divisibility.