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

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

\dots .

n = 12 : m = 20, 21, \dots, 30

Commented by SLVR last updated on 06/Jan/22

W o w \dots 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 \dots 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} + \dots) (1 + x^{3} + x^{6} + \dots) (1 + x^{5} + x^{10} + \dots)

= \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 \dots 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 \dots 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} \dots 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 \dots

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 .

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