Expression, modular arithmetic

[bremenfallturm]

Homework Statement

Solve $15\cdot 16-7(9+10)+11$ in $\mathbb Z_{17}$

Relevant Equations

Given a number $a\equiv b \pmod n$, $a+c\equiv b+c \pmod n$

This is basic modular arithmetic but I just can't get it to work no mather how many different methods I try.
I probably have failed to understand some basics of modular algebra...

Help is appreciated!
Correct is supposed to be $16$

 

[fresh_42]

Taking the remainder respects multiplication and addition. Whenever you get a number greater than $16$ you can reduce it by its remainder. E.g. $15\cdot 16=240=17\cdot 14+2$ so $15\cdot 16 \equiv 2 \pmod{17}$ and the same holds for $9+10$ etc. Negative numbers become positive by e.g. $-14=(-1)\cdot 17 +3 \equiv 3\pmod{17}.$

 

[pasmith]

Homework Statement: Solve $15\cdot 16-7(9+10)+11$ in $\mathbb Z_{17}$
Relevant Equations: Given a number $a\equiv b \pmod n$, $a+c\equiv b+c \pmod n$

This is basic modular arithmetic but I just can't get it to work no mather how many different methods I try.
I probably have failed to understand some basics of modular algebra...
View attachment 351313
Help is appreciated!
Correct is supposed to be $16$

You correctly found $15 \equiv -2$ and $16 \equiv -1$, but when you multiplied these together you somehow got $-2$ instead of $2$. The rest of of your working is correct, but more complicated than necessary. $7 \equiv 7$ and $19 \equiv 2$, so $7 \cdot 19 \equiv 14$; $11 \equiv 11$. Thus $$15 \cdot 16 - 7(10 + 9) + 11 \equiv 2 - 14 + 11 \equiv 2 - 3 \equiv -1 \equiv 16.$$

 

[Mark44]

Homework Statement: Solve $15\cdot 16-7(9+10)+11$ in $\mathbb Z_{17}$

Minor nit -- The above is an expression, so you're not asked to "solve" it, but only to evaluate it. You can solve equations or inequalities that involve unknown variables.

 

[bremenfallturm]

You correctly found $15 \equiv -2$ and $16 \equiv -1$, but when you multiplied these together you somehow got $-2$ instead of $2$. The rest of of your working is correct, but more complicated than necessary. $7 \equiv 7$ and $19 \equiv 2$, so $7 \cdot 19 \equiv 14$; $11 \equiv 11$. Thus $$15 \cdot 16 - 7(10 + 9) + 11 \equiv 2 - 14 + 11 \equiv 2 - 3 \equiv -1 \equiv 16.$$

Thank you! I understand what I did wrong now. Calculating in ##\mathbb Z_n$$ is obviously something you need to get used to :) I'll practise more and ask further questions in a new topic if I need more help!

Minor nit -- The above is an expression, so you're not asked to "solve" it, but only to evaluate it. You can solve equations or inequalities that involve unknown variables.

I see, of course I meant "calculate". Sorry!