26≡R_1 [37] 1 ≡R_2 [3] 2≡R_3 [5] Find R_1 , R_2 and R


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Question Number 90200 by Ar Brandon last updated on 21/Apr/20

$26 \equiv R_{1} [37]$ $1 \equiv R_{2} [3]$ $2 \equiv R_{3} [5]$ $Find R_{1}, R_{2} and R_{3}$
Commented by Rio Michael last updated on 22/Apr/20

$R_{1} = 11, R_{2} = 2, R_{3} = 3$
Commented by Ar Brandon last updated on 22/Apr/20

$Thanks, but how ?$
Commented by Rio Michael last updated on 22/Apr/20

$if your question is actually$ $26 \equiv R_{1} (\mod 37) \Rightarrow R_{1} is the reminder when 26 is divided by 37$ $which rightly is 11 .$ $or you can say$ $by definition a \equiv b (\mod n) \Leftrightarrow n ∣ (a - b) \Rightarrow a - b = n k, k \in Z$ $\Rightarrow 26 \equiv R_{1} (\mod 37) \Rightarrow 26 = 37 k + R$ $k = 0 R_{1} = 26$ $k = 1 \Rightarrow R_{1} = - 11$ $k = - 1 \Rightarrow R_{1} = 63$ $these are just the elements of the set of least residue \mod 37 .$
Commented by Ar Brandon last updated on 22/Apr/20

$Thank you - thank you$
Commented by mr W last updated on 22/Apr/20

$h o w c a n y o u s a y$ $26 \equiv R_{1} m o d 37 \Rightarrow R_{1} = 11 ?$ $t h a t m e a n s t h e n 26 = 37 k + 11 .$ $t h i s i s w r o n g!$ $c l e a r l y :$ $R_{1} = 26 s i n c e 26 \equiv 26 m o d 37$ $R_{2} = 1$ $R_{3} = 2$
Commented by Ar Brandon last updated on 22/Apr/20

$what about - 11, - 2, and - 3 ?$
Commented by mr W last updated on 22/Apr/20

$b u t r e m a i n d e r i s u s u a l l y 0 o r p o s i t i v e$ $n u m b e r .$
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