SOLUTION: Prove that, if d is a factor of x,and d is also a factor of y where d>=1,x>=1,y>=1 ∈ Z then d^2 is a factor of x multiplied by y

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Question 1042995: Prove that, if d is a factor of x,and d is also a factor of y
where d>=1,x>=1,y>=1 ∈ Z
then d^2 is a factor of x multiplied by y
Found 2 solutions by ikleyn, LinnW:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
Prove that, if d is a factor of x,and d is also a factor of y
where d>=1,x>=1,y>=1 ∈ Z
then d^2 is a factor of x multiplied by y
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

From the condition, a = nd  and  b = md  with some integers  n  and  m.

Then ab = nd*md = nm*d^2.


Proved.

And solved.


Answer by LinnW(1048)   (Show Source): You can put this solution on YOUR website!
Since d is a factor of x, there exists a positive integer e
such that x = d*e .
Since d is a factor of y, there exists a positive integer f
such that y = d*f .
So x*y = (d*e)(d*f)
Since order is not important in multiplication
x*y = (d*d*e*f) or x*y = d^2*e*f
and d^2 is a factor of x*y
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