Expression, modular arithmetic

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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...
1726766286268.png

Help is appreciated!
Correct is supposed to be ##16##
 
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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}.##
 
bremenfallturm said:
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 <br /> 15 \cdot 16 - 7(10 + 9) + 11 \equiv 2 - 14 + 11 \equiv 2 - 3 \equiv -1 \equiv 16.
 
bremenfallturm said:
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.
 
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pasmith said:
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 <br /> 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!
Mark44 said:
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!
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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