{: (( a^2 +b^2 =c^2 )),((a+b+c=1000)) }; a^(?) , b^(?) , c^(?) ∈Z Q#167533 reposted


Question and Answers Forum

All Questions Topic List
Number Theory Questions
Previous in All Question Next in All Question
Previous in Number Theory Next in Number Theory
Question Number 167650 by Rasheed.Sindhi last updated on 22/Mar/22
${: (( a^2 +b^2 =c^2 )),((a+b+c=1000)) }; a^(?) , b^(?) , c^(?) ∈Z Q#167533 reposted$
$\begin{matrix} a^{2} + b^{2} = c^{2} \\ a + b + c = 1000 \end{matrix}}; \overset{?}{a}, \overset{?}{b}, \overset{?}{c} \in Z$ $You can't use 'macro parameter character #' in math mode$
Commented by Ari last updated on 22/Mar/22

Commented by mr W last updated on 22/Mar/22

$i g o t o n l y o n e s o l u t i o n f o r a, b, c \in N$ $(200, 375, 425)$
Commented by Rasheed.Sindhi last updated on 22/Mar/22

$B u t s i r f o r a, b, c \in Z t h e r e a r e s e v e r a l$ $s o l u t i o n s, I t h i n k .$
Commented by Rasheed.Sindhi last updated on 22/Mar/22

$a, b, c a r e i n t e g e r s .$
	Answered by Rasheed.Sindhi last updated on 22/Mar/22
	$determinant ((( { ((a=m^2 −n^2 )),((b=2mn)),((c=m^2 +n^2 )) :}∀ m,n∈Z}⇒a^2 +b^2 =c^2 ))) AND determinant (((a^2 +b^2 =c^2 ⇒ { ((a=m^2 −n^2 )),((b=2mn)),((c=m^2 +n^2 )) :} for some m,n∈Z))) a+b+c =(m^2 −n^2 )+(2mn)+(m^2 +n^2 )=1000 ⇒2m^2 +2mn=1000 m^2 +mn=500 n=((500−m^2 )/m)=((500)/m)−m ∵n∈Z ∴ m∣500 m=±1,±2,±4,±5,±10,±20,±25,±50,±100,±125,±250,±500 { ((a=m^2 −n^2 =m^2 −(((500)/m)−m)^2 )),((b=2mn=2m(((500)/m)−m))),((c=m^2 +n^2 =m^2 +(((500)/m)−m)^2 )) :} OR { ((a=2mn=2m(((500)/m)−m))),((b=m^2 −n^2 =m^2 −(((500)/m)−m)^2 )),((c=m^2 +n^2 =m^2 +(((500)/m)−m)^2 )) :} Where m is positive/negative divisor of 500. determinant ((m,n_(=((500)/m)−m) ,a_(=m^2 −n^2 _(=2mn) ) ,b_(=2mn_(=m^2 −n^2 ) ) ,c_(=m^2 +n^2 ) ),((±1),(±499),(−249000_(998) ),(998_(−249000) ),(249002_(249002) )),((±2),(±248),(−61500_(992) ),(992_(−61500) ),(61508_(61508) )),((±4),(±121),(−14625_(968) ),(968_(−14625) ),(14657_(14657) )),((±5),(±95),(−9000_(950) ),(950_(−9000) ),(9050_(9050) )),((±10),(±40),(−1500_(800) ),(800_(−1500) ),(1700_(1700) )),((±20),(±5),(375_(200) ),(200_(375) ),(425_(425) )),((±25),(∓5),(600_(−250) ),(−250_(600) ),(650_(650) )),((±50),(∓40),(900_(−4000) ),(−4000_(900) ),(4100_(4100) )),((±100),(∓95),(975_(−19000) ),(−19000_(975) ),(19025_(19025) )),((±125),(∓121),(984_(−30250) ),(−30250_(984) ),(30266_(30266) )),((±250),(∓248),(996_(−124000) ),(−124000_(996) ),(124004_(124004) )),((±500),(∓499),(999_(−499000) ),(−499000_(999) ),(499001_(499001) ))) 24 solutions$
	$\begin{matrix} {\begin{cases} a = m^{2} - n^{2} \\ b = 2 m n \\ c = m^{2} + n^{2} \end{cases} \forall m, n \in Z} \Rightarrow a^{2} + b^{2} = c^{2} \end{matrix}$ $AND$ $\begin{matrix} a^{2} + b^{2} = c^{2} \Rightarrow {\begin{cases} a = m^{2} - n^{2} \\ b = 2 m n \\ c = m^{2} + n^{2} \end{cases} f o r s o m e m, n \in Z \end{matrix}$ $a + b + c$ $= (m^{2} - n^{2}) + (2 m n) + (m^{2} + n^{2}) = 1000$ $\Rightarrow 2 m^{2} + 2 m n = 1000$ $m^{2} + m n = 500$ $n = \frac{500 - m^{2}}{m} = \frac{500}{m} - m$ $∵ n \in Z ∴ m ∣ 500$ $m = \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm 25, \pm 50, \pm 100, \pm 125, \pm 250, \pm 500$ ${\begin{cases} a = m^{2} - n^{2} = m^{2} - {(\frac{500}{m} - m)}^{2} \\ b = 2 m n = 2 m (\frac{500}{m} - m) \\ c = m^{2} + n^{2} = m^{2} + {(\frac{500}{m} - m)}^{2} \end{cases}$ $O R$ ${\begin{cases} a = 2 m n = 2 m (\frac{500}{m} - m) \\ b = m^{2} - n^{2} = m^{2} - {(\frac{500}{m} - m)}^{2} \\ c = m^{2} + n^{2} = m^{2} + {(\frac{500}{m} - m)}^{2} \end{cases}$ $W h e r e m i s p o s i t i v e / n e g a t i v e$ $d i v i s o r o f 500 .$ $\begin{array}{ccccccccccccc} m & \underset{= \frac{500}{m} - m}{n} & \underset{= \underset{= 2 m n}{m^{2} - n^{2}}}{a} & \underset{\underset{= m^{2} - n^{2}}{= 2 m n}}{b} & \underset{= m^{2} + n^{2}}{c} \\ \pm 1 & \pm 499 & \underset{998}{- 249000} & \underset{- 249000}{998} & \underset{249002}{249002} \\ \pm 2 & \pm 248 & \underset{992}{- 61500} & \underset{- 61500}{992} & \underset{61508}{61508} \\ \pm 4 & \pm 121 & \underset{968}{- 14625} & \underset{- 14625}{968} & \underset{14657}{14657} \\ \pm 5 & \pm 95 & \underset{950}{- 9000} & \underset{- 9000}{950} & \underset{9050}{9050} \\ \pm 10 & \pm 40 & \underset{800}{- 1500} & \underset{- 1500}{800} & \underset{1700}{1700} \\ \pm 20 & \pm 5 & \underset{200}{375} & \underset{375}{200} & \underset{425}{425} \\ \pm 25 & \mp 5 & \underset{- 250}{600} & \underset{600}{- 250} & \underset{650}{650} \\ \pm 50 & \mp 40 & \underset{- 4000}{900} & \underset{900}{- 4000} & \underset{4100}{4100} \\ \pm 100 & \mp 95 & \underset{- 19000}{975} & \underset{975}{- 19000} & \underset{19025}{19025} \\ \pm 125 & \mp 121 & \underset{- 30250}{984} & \underset{984}{- 30250} & \underset{30266}{30266} \\ \pm 250 & \mp 248 & \underset{- 124000}{996} & - \underset{996}{124000} & \underset{124004}{124004} \\ \pm 500 & \mp 499 & \underset{- 499000}{999} & \underset{999}{- 499000} & \underset{499001}{499001} \end{array}$ $24 solutions$
	Commented by mr W last updated on 22/Mar/22

	$n i c e s o l u t i o n!$
	Commented by Rasheed.Sindhi last updated on 22/Mar/22

	$\begin{matrix} T han k s sir! \end{matrix}$
	Commented by Tawa11 last updated on 22/Mar/22

	$Great sir$
	Commented by Rasheed.Sindhi last updated on 23/Mar/22

	$\begin{matrix} \underset{M i s s}{TH α n k S} \end{matrix}$
	Commented by Rasheed.Sindhi last updated on 26/Mar/22

	$T w o s o l u t i o n s w h i c h a r e n o t c o v e r e d$ $b y a b o v e a p p r o a c h a r e s h a r e d b y m y$ $n e p h e w F e r o z (0, 500, 500) a n d$ $(500, 0, 500) .$
	Commented by Rasheed.Sindhi last updated on 29/Mar/22
	${: (( a^2 +b^2 =c^2 )),((a+b+c=1000)) }; a^(?) , b^(?) , c^(?) ∈Z c^2 −b^2 =a^2 ⇒(c−b)(c+b)=a^2 A Case:c−b=c+b=a −b=b⇒b=0 { ((a^2 +b^2 =c^2 )),((a+b+c=1000)) :}⇒ { ((a^2 =c^2 )),((a+c=1000⇒a=1000−c)) :} a^2 =c^2 ⇒(1000−c)^2 =c^2 10^6 −2000c+c^2 =c^2 2000c=10^6 2c=1000 c=500 a=1000−c=1000−500=500 (a,b,c)=(500,0,500) Similarly (a,b,c)=(0,500,500)$
	$\begin{matrix} a^{2} + b^{2} = c^{2} \\ a + b + c = 1000 \end{matrix}}; \overset{?}{a}, \overset{?}{b}, \overset{?}{c} \in Z$ $c^{2} - b^{2} = a^{2} \Rightarrow (c - b) (c + b) = a^{2}$ $A Case : c - b = c + b = a$ $- b = b \Rightarrow b = 0$ ${\begin{cases} a^{2} + b^{2} = c^{2} \\ a + b + c = 1000 \end{cases} \Rightarrow {\begin{cases} a^{2} = c^{2} \\ a + c = 1000 \Rightarrow a = 1000 - c \end{cases}$ $a^{2} = c^{2} \Rightarrow {(1000 - c)}^{2} = c^{2}$ $10^{6} - 2000 c + c^{2} = c^{2}$ $2000 c = 10^{6}$ $2 c = 1000$ $c = 500$ $a = 1000 - c = 1000 - 500 = 500$ $(a, b, c) = (500, 0, 500)$ $S i m i l a r l y (a, b, c) = (0, 500, 500)$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum