An (almost) impossible modular arithmetic question

[theax]

\(\displaystyle (a + b) \pmod{m} \equiv a \pmod{m} + b \pmod{m}\)
\(\displaystyle ab \pmod{m} \equiv (a \pmod{m} )( b \pmod{m})\)

Can anyone help me prove these properties?????????

 

[stapel]

\(\displaystyle (a + b) \pmod{m} \equiv a \pmod{m} + b \pmod{m}\)
\(\displaystyle ab \pmod{m} \equiv (a \pmod{m} )( b \pmod{m})\)

First, review the definition of what it means for some number x to be equal to another number y, "modulo m".

Then try stating "a" and "b" in terms of their modular definitions. If a = p + qm and b = s + tm, then what is the value of (a + b)-mod-m? What is the value of the sum of a-mod-m and b-mod-m? Can you show equality?

Follow the same process with (a*b)-mod-m. :wink:

Eliz.

 

[theax]

Thanks very much!!!!!!!!!!!!!!

I'm kind of new to modular arithmetic and I tried my many many ways of proving this and I failed.
Can someone write out step by step how you would go about proving this property?

Any insight into this problem is greatly appreciated and I thank you in advance.

 

[mmm4444bot]

HUH?

… Can someone write out step by step how you would go about proving this property?

Any insight into this problem is greatly appreciated …

The steps posted by Elizabeth sure look like steps for how to "go about" starting the proof, to me.

You posted your "thank you" three minutes after Elizabeth posted these steps.

What fractional part of that three minutes did you spend thinking about these steps? (This is a rhetorical question.)

 

[theax]

One can actually do a lot in three minutes. I substituted the a = p + qm and b = s + tm in on the right side of the equation, but I could not substitute the value into the left side of the equation.

Can anyone help me with this problem? Any help is greatly appreciated and I thank you in advance.

 

[stapel]

One can actually do a lot in three minutes. I substituted the a = p + qm and b = s + tm in on the right side of the equation, but....

Okay, so you didn't review the definition, and you copied down what you were given. That took, what?, about thirty seconds...?

Now please go and review the definition. Express the other side of the equation according to the definition, do the actual addition "on the right side of the equation", and try to figure out how to apply the definition to the simplified result of that addition, comparing the two sides of the equation and using the definition to show equality.

If you get stuck, please reply showing your work and reasoning so far. Thank you.

Eliz.