Find the non negative integer solutions of 2x+3y+5z=60


Question and Answers Forum

All Questions Topic List
Permutation and Combination Questions
Previous in All Question Next in All Question
Previous in Permutation and Combination Next in Permutation and Combination
Question Number 163313 by SLVR last updated on 06/Jan/22

$F i n d t h e n o n n e g a t i v e i n t e g e r$ $s o l u t i o n s o f 2 x + 3 y + 5 z = 60$
	Answered by mr W last updated on 06/Jan/22

	$l e t 2 x + 3 y = 5 u$ $5 u + 5 z = 60$ $u + z = 12$ $\Rightarrow u = n w i t h n \in Z$ $\Rightarrow z = 12 - n$ $2 x + 3 y = 1$ $x = 3 k - 1 w i t h k \in Z$ $y = - 2 k + 1$ $2 x + 3 y = 5 u = 5 n$ $x = 5 n (3 k - 1) = 3 (5 n k) - 5 n = 3 m - 5 n$ $y = 5 n (- 2 k + 1) = - 2 (5 n k) + 5 n = - 2 m + 5 n$ $g e n e r a l s o l u t i o n :$ ${\begin{cases} x = 3 m - 5 n \\ y = - 2 m + 5 n \\ z = 12 - n \end{cases} (w i t h m, n \in Z)$ $n o n - n e g a t i v e s o l u t i o n s :$ $z = 12 - n ⩾ 0 \Rightarrow n ⩽ 12$ $x = 3 m - 5 n ⩾ 0$ $y = - 2 m + 5 n ⩾ 0$ $\Rightarrow 0 ⩽ n ⩽ 12$ $\Rightarrow \frac{5 n}{3} ⩽ m ⩽ \frac{5 n}{2}$ $t o t a l l y t h e r e a r e 71 s o l u t i o n s :$ $n = 0 : m = 0$ $n = 1 : m = 2$ $n = 2 : m = 3, 4$ $n = 3 : m = 5, 6, 7$ $. . . .$ $n = 12 : m = 20, 21, . . ., 30$
	Commented by SLVR last updated on 06/Jan/22

	$W o w . . . r e a l l y g r e a t e n o u g h . . w e$ $a r e b l e s s e d . . . w i t h y o u r s e r v i c e$
	Commented by mr W last updated on 06/Jan/22

	Commented by mr W last updated on 06/Jan/22

	$\underset{―}{n o t e :}$ $n u m b e r o f n o n n e g a t i v e s o l u t i o n s o f$ $2 x + 3 y + 5 z = 60 i s t h e c o e f . o f t e r m x^{60}$ $i n t h e e x p a n s i o n o f$ $(1 + x^{2} + x^{4} + . . .) (1 + x^{3} + x^{6} + . . .) (1 + x^{5} + x^{10} + . . .)$ $= \frac{1}{(1 - x^{2}) (1 - x^{3}) (1 - x^{5})} . t h a t i s 71 .$
	Commented by mr W last updated on 06/Jan/22

	Commented by Rasheed.Sindhi last updated on 06/Jan/22

	$\underset{\underset{P \in RF \in\subset T!}{THAN}}{M RE}$ $M ultimedia (graph)$ $& number of solutions$ $are in addition!$ $T h a n X s i r!$
	Commented by mr W last updated on 06/Jan/22

	$t h a n k s s i r s!$ $g r a p h h e l p s v e r y m u c h t o f i n d o u t i f$ $e r r o r s a r e m a d e . t h e r e f o r e i l i k e t o$ $w o r k w i t h g r a p h .$
	Commented by Tawa11 last updated on 06/Jan/22

	$Great sir$
	Commented by Rasheed.Sindhi last updated on 07/Jan/22
	SHARING: https://youtu.be/fw1kRz83Fj0
	Commented by SLVR last updated on 31/Jan/22

	$R e s p e c t e d p r o f . W . . . e v a l u a t i o n$ $o f x^{60} w i t h o u t o r i g i n a l m u l$ $t l i c a t i o n a l l t e r m s c o m p l n e n t$ $w i s e . . k i n d l y w i t h g e n e r a l$ $t e r m o f e x p a n s i o n . p l e a s e$
	Commented by mr W last updated on 31/Jan/22

	$p l e a s e e x p r e s s c l e a r l y w h a t y o u w a n t$ $t o h a v e!$
	Commented by SLVR last updated on 02/Feb/22

	$S i r . . . I m e a n t o s a y . .$ $c o e f f i t i e n t o f x^{60} i n (1 - x^{2}) (1 - x^{3}) (1 - x^{5})$ $n o t b y m u l t i p l y i n g a l l t e r m s$ $b u t p o s s i b i l i t y o f g e n e r a l t e r m o f {(1 - x)}^{- 1}$ $C_{r}^{n + r - 1} . . . O r g e t t i n g t h e c o e f f i t i e n t$ $o f x^{60} a s 71 i n a e a s y w a y ? ? ?$ $P l e a s e . . s i r . . .$
	Commented by mr W last updated on 02/Feb/22

	$i n f a c t t h e r e i s n o e a s y w a y t o$ $d e t e r m i n e t h e c o e f f i c i e n t o f a$ $g e n e r a l t e r m .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum