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 January 31st, 2018, 05:59 PM #1 Newbie   Joined: Jan 2018 From: Toronto Posts: 12 Thanks: 0 Mod Opertaor I don't understand this question - Please help me understand why it is true or false. If a â‰¡ x1 (mod m) and b â‰¡ y1 (mod m), then we will have ab â‰¡ x1y1 (mod m). True or False and why? Thank you.
 January 31st, 2018, 06:05 PM #2 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,749 Thanks: 613 Math Focus: Yet to find out. Did you try play around with some examples? Suppose $m = 2\pi$ x1, y1 are angles for example. What happens when you consider different values for x1 and y1 for the ab expression?
January 31st, 2018, 06:12 PM   #3
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 Originally Posted by Joppy Did you try play around with some examples? Suppose $m = 2\pi$ x1, y1 are angles for example. What happens when you consider different values for x1 and y1 for the ab expression?
I have not tried diffrent values. Not understanding what a or x1, y1 would represent. In addition to what numeric m (mod m) would represent.

I understand when written out like this 33 mod 8, how to get the results, but the above question is really not clear.

January 31st, 2018, 06:34 PM   #4
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 Originally Posted by Tricia If a â‰¡ x1 (mod m) and b â‰¡ y1 (mod m), then we will have ab â‰¡ x1y1 (mod m). True or False and why?
Here's a complete solution, although playing around with Joppy's hint would be valuable too so you can see more examples of how this works.

The trick to this type of problem is to break it down to what it's really saying. To do that, we have to get picky with the definitions.

If $a = x_1 \pmod m$ then $m | a - x_1$ (The vertical bar means "divides")

Then (by the definition of divides) there is some integer $d_a$ (d for divisor) such that $m d_a = a - x_1$, or

(1) $a = m d_a + x_1$

And likewise there is some integer $d_b$ such that

(2) $b = m d_b + x_2$

Now using (1) and (2) we have

$a b = (m d_a + x_1)(m d_b + x_2)$

$= m^2 d_a d_b + m d_a x_2 + m d_b x_1 + x_1 x_2$

$= m(m d_a d_b + d_a x_2 + d_b x_1) + x_1 x_2$

Now you can see that $a b - x_1 x_2$ is a multiple of $m$ and we're done.

This is how all of these kinds of problems should be approached. Replace each technical term or symbol you are learning about with its precise definition. Whenever you get stuck, just go symbol-by-symbol and ask yourself: Exactly what does this mean? Write down the textbook definition. After that the problem often solves itself. This is one of the standard patterns for doing problems. Always make sure you understand exactly what the definitions say.

Last edited by Maschke; January 31st, 2018 at 06:54 PM.

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