How many triples of positive integers (x,y,z) satisfy 79x+80y+81z =2016


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Question Number 111730 by Aina Samuel Temidayo last updated on 04/Sep/20

$How many triples of positive integers$ $(x, y, z) satisfy$ $79 x + 80 y + 81 z = 2016$
Commented by kaivan.ahmadi last updated on 05/Sep/20

$l e t y = t b e f i x$ $79 x + 81 z = 2016 - 80 t ()$ $79 x \overset{81}{\equiv} - 9 + t \Rightarrow - 2 x \overset{81}{\equiv} - 9 + t \overset{\times (- 41)}{\Rightarrow}$ $82 x \overset{81}{\equiv} 369 - 41 t \Rightarrow x \overset{81}{\equiv} 45 - 41 t \Rightarrow$ $x = 81 k - 41 t + 45 (k \in Z) - 79$ $f r o m ()$ $\Rightarrow 79 (81 k + 45 - 41 t) + 81 z = 2016 - 80 t \Rightarrow$ $81 z = 81 (- 79 k) + 3159 t - 1539 \Rightarrow$ $z = - 79 k + 39 t - 19$ $x, y, z \in N \Rightarrow 81 k - 41 t + 45 > 0, - 79 k + 39 t - 19 > 0$ $\Rightarrow \frac{41 t - 45}{81} < k < \frac{39 t - 19}{79}$ $i f t = 1 \Rightarrow \frac{- 4}{81} < k < \frac{20}{79} \Rightarrow k = 0 \Rightarrow$ $x = 4, y = 1, z = 20$ $i f t = 2 \Rightarrow \frac{37}{81} < k < \frac{59}{79} \Rightarrow t h e r e i s n o k$ $i f t = 3 \Rightarrow \frac{78}{81} < k < \frac{98}{79} \Rightarrow k = 1 \Rightarrow$ $x = 3, y = 3, z = 19$ $i f t = 4 \Rightarrow \frac{119}{81} < k < \frac{137}{79} \Rightarrow n o k$ $i f t = 5 \Rightarrow \frac{160}{81} < k < \frac{176}{79} \Rightarrow k = 2 \Rightarrow$ $x = 2, y = 5, z = 18$ $a n d r e p e a t t h i s r u l e w e g e t a n o t h e r a n s w e r s .$
Commented by kaivan.ahmadi last updated on 05/Sep/20

$T h a n k y o u s i r . Y o u a r e r i g h t .$ $I c h a n g e d t h e a n s w e r .$
Commented by kaivan.ahmadi last updated on 05/Sep/20

$y o u c a n p l u s e o r m i n e s t h e m u l t i p l e o f 81$ $t o a n y s i d e o f \equiv$
Commented by Aina Samuel Temidayo last updated on 05/Sep/20

$79 x \overset{81}{\equiv} - 9 + t \Rightarrow - 2 x \overset{81}{\equiv} - 9 + t$ $Pleas what do you mean by this ?$
Commented by kaivan.ahmadi last updated on 05/Sep/20

$79 x - 81 x \overset{81}{\equiv} - 9 + t$
Commented by Rasheed.Sindhi last updated on 05/Sep/20

$@ k i v a n a h m a d i$ $(1, 7, 17) a l s o s a t i s f y .$ $P l e a s e r e v i e w y o u r a n s w e r t h e r e$ $m i g h t b e o t h e r s o l u t i o n s a l s o .$
Commented by Aina Samuel Temidayo last updated on 05/Sep/20

$Why is 81 still on top of the$ $equivalent sign ? Please I {don}^{'} t get .$
Commented by Aina Samuel Temidayo last updated on 05/Sep/20

$Oh . Ok . Thanks .$
Commented by Rasheed.Sindhi last updated on 05/Sep/20

$@ A i n a$ $S i r A h m a d i h a s w r i t t e n$ $79 x - 81 x \overset{81}{\equiv} - 9 + t$ $i n s t e a d o f$ $79 x - 81 x \equiv - 9 + t (m o d 81)$
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