# consider the relation division a|b. prove that a|b is transitive then if `a,b>0` and `a|b` then `a<=b` .

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By definition `a|b` if and only if `b=an,` `n in ZZ`.

A relation `circ` is transitive if `(a circ b ^^b circ c)=>(a circ c)`. So we need to prove that if `a|b` and `b|c` then `a|c`.

If `a|b` then `b=an`

If `b|c` then `c= bm=anm`

Since `m` and `n` are integers then `mn` is also integer which means that `a|b.`

Let `a,b>0` and `a|b`.

Since `a|b` it follows that `b=an` for some integer `n` and since `b` is positive `an` must be positive and since `a` is positive `n` must be positive as well. So `n >=1`. If `n=1` then `a=b`.

If `n>1` then `a=b/n` and since both `b` and `n` are positive `b/n<b`.

Hence `a<=b`

Let a divides b i.e. `b/a ,b>=a ,b= ma ,`

and m is an integer.

Let b divides c i.e `c/b ,c>=b ,c=nb ,`

and n is an integer.

c=n(ma)=(nm)a ,mn is an integer , this mean a is multiple of c

i.e. a divides c i.e a/c

this proves transitivity.