"Prove that if x and y are integers such that x ≡ 6 (mod 8)...

[nikosan]

"Prove that if x and y are integers such that x ≡ 6 (mod 8) and y ≡ 3 (mod 8), then 8 divides 2(x + y) 21 + 3xy."

how do I approach a question like this? i'm studying for a discrete maths exam on tuesday. thanks

 

[stapel]

"Prove that if x and y are integers such that x ≡ 6 (mod 8) and y ≡ 3 (mod 8), then 8 divides 2(x + y) 21 + 3xy."

The formatting in the last expression is quite non-standard. What is meant by the "21" that follows the "(x + y)"?

how do I approach a question like this?

What methods did they cover in class? What results (theorems, rules, etc) have you been given to use?

Please be complete. Thank you!

 

[nikosan]

The formatting in the last expression is quite non-standard. What is meant by the "21" that follows the "(x + y)"?


What methods did they cover in class? What results (theorems, rules, etc) have you been given to use?

Please be complete. Thank you!

sorry, I copy and pasted it without realising it wouldn't come out properly. it should read ^21.

we haven't really been given specific formulas or theories for questions like this. another question we got was "Is there an integer z such that 255z is congruent to 7 (mod 633), and we work it out by 255 = 85 x 3, and 633 = 3 x 211, so z must be a multiple of 3 which 7 can't be. this question seems to be about applying similar concepts

 

[stapel]

sorry, I copy and pasted it without realising it wouldn't come out properly. it should read ^21.

So the question is as follows:

. . . . .\(\displaystyle \mbox{Prove that, if }\, x\, \mbox{ and }\, y\, \mbox{ are integers such that}\)

. . . . .\(\displaystyle x\, \equiv\, 6\, \mbox{(mod 8) and }\, y\, \equiv\, 3\, \mbox{(mod 8), then }\, \)

. . . . .\(\displaystyle 8\, \mbox{ divides } \, 2\, (x\, +\, y)^{21}\, +\, 3xy\)

we haven't really been given specific formulas or theories for questions like this.

Then I guess you'll just have to work from the definition of congruency, mod 8, and create expressions which will allow you to show the result.

. . . . .x = 8m + 6

. . . . .y = 8n + 3

...and so forth. The result does follow.