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Question Number 68537 by naka3546 last updated on 13/Sep/19

Answered by MJS last updated on 13/Sep/19

$$1817$$$$4717$$$$41313$$$$7716$$$$81113$$

Commented by naka3546 last updated on 13/Sep/19

$$pleaseshowyourworkings$$

Commented by MJS last updated on 13/Sep/19

$$\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{try}$$$$\sqrt{354-a^2-b^2}=c$$$$\mathrm{let}a⩽b⩽c$$$$\mathrm{we}\mathrm{find}a⩽\lfloor \sqrt{354}\rfloor \wedge a⩽b⩽\lfloor \sqrt{177-\frac{a^2}{2}}\rfloor \Rightarrow$$$$\Rightarrow 1⩽a⩽10\wedge a⩽b⩽\lfloor \sqrt{177-\frac{a^2}{2}}\rfloor$$$$\mathrm{this}\mathrm{leads}\mathrm{to}\mathrm{the}\mathrm{following}\mathrm{borders}:$$$$1⩽a⩽4\Rightarrow a⩽b⩽13$$$$5⩽a⩽8\Rightarrow a⩽b⩽12$$$$9⩽a⩽10\Rightarrow a⩽b⩽11$$

Answered by Rasheed.Sindhi last updated on 18/Sep/19

$$\mathbb{A}\mathbb{D}\mathbf{ifferent}\mathbb{W}\mathbf{ay}$$$$\mathrm{Can}\mathrm{be}\mathrm{proved}\mathrm{that}\mathrm{exactly}\mathbf{one}\mathrm{of}$$$$\mathrm{a},\mathrm{b}\mathrm{\&}\mathrm{c}\mathrm{is}{\mathrm{even}}^{★}\mathrm{Also}1⩽\mathrm{a},\mathrm{b},\mathrm{c}⩽18.$$$$\mathrm{Assume}\mathrm{that}\mathrm{a}\in \mathbb{E}\mathrm{\&}\mathrm{b},\mathrm{c}\in \mathbb{O}$$$$\mathrm{let}\mathrm{a}=2\mathrm{u},\mathrm{b}=2\mathrm{v}+1,\mathrm{c}=2\mathrm{w}+1$$$${\left(2\mathrm{u}\right)}^2+{(2\mathrm{v}+1)}^2+{(2\mathrm{w}+1)}^2=354$$$${4\mathrm{u}}^2+{4\mathrm{v}}^2+4\mathrm{v}+1+{4\mathrm{w}}^2+4\mathrm{w}+1=354$$$${4\mathrm{u}}^2+{4\mathrm{v}}^2+4\mathrm{v}+{4\mathrm{w}}^2+4\mathrm{w}=352$$$${\mathrm{u}}^2+{\mathrm{v}}^2+\mathrm{v}+{\mathrm{w}}^2+\mathrm{w}=88$$$${\mathrm{u}}^2+\mathrm{v}(\mathrm{v}+1)+\mathrm{w}(\mathrm{w}+1)=88$$$$\mathrm{v}(\mathrm{v}+1)\in \mathbb{E}\wedge \mathrm{w}(\mathrm{w}+1)\in \mathbb{E}$$$$\Rightarrow {\mathrm{u}}^2\in \mathbb{E}\Rightarrow \mathrm{u}\in \mathbb{E}\Rightarrow \mathrm{a}=2\mathrm{u}\mathrm{is}\mathrm{doubly}\mathrm{even}.$$$$\mathrm{Possible}\mathrm{values}\mathrm{for}\mathrm{a}:4,8,12\mathrm{\&}16$$$$\mathrm{For}\mathrm{a}=4:\mathrm{c}=\sqrt{354-{\mathrm{a}}^2-{\mathrm{b}}^2}=\sqrt{338-{\mathrm{b}}^2}$$$$\mathrm{Possible}\mathrm{values}\mathrm{for}\mathrm{b}:1,3,5,...,17$$$$\mathrm{Only}\mathrm{for}\mathrm{b}=7,13\mathrm{\&}17;\mathrm{c}\in {\mathbb{Z}}^{+}$$$$\mathrm{Successful}\mathrm{triplets}:4,7,17\mathrm{\&}4,13,13$$$$\mathrm{For}\mathrm{a}=8:\mathrm{c}=\sqrt{354-{\mathrm{a}}^2-{\mathrm{b}}^2}=\sqrt{290-{\mathrm{b}}^2}$$$$\mathrm{Possible}\mathrm{values}\mathrm{for}\mathrm{b}:1,3,5,...,17$$$$\mathrm{Only}\mathrm{for}\mathrm{b}=1,11,13\mathrm{\&}17;\mathrm{c}\in {\mathbb{Z}}^{+}$$$$\mathrm{Successful}\mathrm{triplets}:8,1,17\mathrm{\&}8,11,13$$$$\mathrm{For}\mathrm{a}=12:\mathrm{c}=\sqrt{354-{\mathrm{a}}^2-{\mathrm{b}}^2}=\sqrt{210-{\mathrm{b}}^2}$$$$\mathrm{Possible}\mathrm{values}\mathrm{for}\mathrm{b}:1,3,5,...,13$$$$\mathrm{For}\mathbf{no}\mathrm{value}\mathrm{of}\mathrm{b};\mathrm{c}\in {\mathbb{Z}}^{+}$$$$\mathrm{No}\mathrm{Successful}\mathrm{triplets}.$$$$\mathrm{For}\mathrm{a}=16:\mathrm{c}=\sqrt{354-{\mathrm{a}}^2-{\mathrm{b}}^2}=\sqrt{98-{\mathrm{b}}^2}$$$$\mathrm{Possible}\mathrm{values}\mathrm{for}\mathrm{b}:1,3,5,...,9$$$$\mathrm{Only}\mathrm{for}\mathrm{b}=7;\mathrm{c}\in {\mathbb{Z}}^{+}$$$$\mathrm{Successful}\mathrm{triplets}:16,7,7$$$$$$

Commented by Rasheed.Sindhi last updated on 14/Sep/19

$${}^{★}{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=354\Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\in \mathbb{E}$$$${}^{\bullet }\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{all}\mathrm{are}\mathrm{not}\mathrm{odd}.\mathrm{If}\mathrm{they}\mathrm{were}$$$${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\mathrm{be}\mathrm{surely}\mathrm{odd}.$$$${}^{\bullet }\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{all}\mathrm{are}\mathrm{not}\mathrm{even}:\mathrm{If}\mathrm{they}\mathrm{were}$$$${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\mathrm{were}\mathrm{doubly}\mathrm{even}\mathrm{but}$$$$\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{because}\mathrm{it}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{a}\mathrm{singly}$$$$\mathrm{even}\left(354\right)$$$$[{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2={\left(2\mathrm{u}\right)}^2+{\left(2\mathrm{v}\right)}^2+{\left(2\mathrm{w}\right)}^2$$$$=4({\mathrm{u}}^2+{\mathrm{v}}^2+{\mathrm{w}}^2)]$$$${}^{\bullet }\mathrm{Two}\mathrm{of}\mathrm{a},\mathrm{b},\mathrm{c}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{be}\mathrm{even}.\mathrm{If}\mathrm{they}$$$$\mathrm{were},{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\mathrm{be}\mathrm{odd}\mathrm{due}\mathrm{to}\mathrm{one}$$$$\mathrm{odd}.$$$${}^{\bullet }\therefore \mathrm{only}\mathrm{one}\mathrm{of}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{is}\mathrm{even}.$$

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