Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 195194 by Abdullahrussell last updated on 26/Jul/23

Answered by Frix last updated on 26/Jul/23

$$abcd$$$$231510$$$$24126$$$$26124$$$$210153$$$$321015$$$$3366$$$$3663$$$$315102$$$$42612$$$$4444$$$$41262$$$$62412$$$$6336$$$$6633$$$$61242$$$$102315$$$$101532$$$$12426$$$$12624$$$$153210$$$$151023$$

Answered by deleteduser1 last updated on 27/Jul/23

$$WLOG,letabethemaximum$$$$ac=a+b+c+d⩽4a\Rightarrow c⩽4;c=1\Rightarrow b+d=-1\left(absurd\right)$$$$c=2\Rightarrow b+d=a-2and2a=bd$$$$banddsatisfy{x}^2-(a-2)x+2a=0$$$$b,d=\frac{(a-2)\underset{-}{+}\sqrt{a^2-12a+4}}{2}\in {\mathbb{Z}}^{+}\Rightarrow a^2-12a+4=p^2$$$${(a-6)}^2-32=p^2\Rightarrow (a-6-p)(a-6+p)=32=1\times 32$$$$=2\times 16=3\times 8...(uptopermutation,negativefactors)$$$${\mathbb{Z}}^{+}solutions:(a,p)=(12,2),(15,7)$$$$\Rightarrow b,d=\frac{10\underset{-}{+}2}{2}=6or4OR\frac{13\underset{-}{+}7}{2}=10or3$$$$c=3\Rightarrow b+d=2a-3and3a=bd$$$$\Rightarrow b,d=\frac{(2a-3)\underset{-}{+}\sqrt{4a^2-24a+9}}{2}$$$$\Rightarrow {(2a-6)}^2-27=p^2\Rightarrow (2a-6-p)(2a-6+p)=27$$$$\Rightarrow a=6,p=3\Rightarrow b,d=\frac{9\underset{-}{+}3}{2}=6or3$$$$Forc=4\Rightarrow b+d=3a-4,bd=4a$$$$\Rightarrow b,d=\frac{(3a-4)\underset{-}{+}\sqrt{9a^2-40a+16={(3a-\frac{20}{3})}^2-{\left(\frac{16}{3}\right)}^2}}{2},$$$$\Rightarrow (3a-\frac{20}{3}-p)(3a-\frac{20}{3}+p)=\frac{256}{9}\Rightarrow a=4,p=0$$$$\Rightarrow b,d=4,4$$$$\Rightarrow (a,b,c,d)=(12,6,2,4),(15,10,2,3),(6,6,3,3),$$$$(4,4,4,4)uptopermutationsofbandd.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com