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Question Number 25619 by aplus last updated on 12/Dec/17

Answered by prakash jain last updated on 12/Dec/17

$$ab+cd=38\left(i\right)$$$$ac+bd=34\left(ii\right)$$$$ad+bc=43\left(iii\right)$$$$add\left(i\right)and\left(ii\right)$$$$a(b+c)+d(b+c)=72$$$$(a+d)(b+c)=72$$$$\Rightarrow (a+d),(b+c)\mathrm{are}\mathrm{factor}\mathrm{of}72.$$$$=1\times 72,2\times 36,3\times 24,4\times 18,6\times 12,8\times 9\left(A\right)$$$$\mathrm{add}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{iii}\right)$$$$a(b+d)+c(b+d)=81$$$$(a+c)(b+d)=81$$$$=1\times 81,3\times 27,9\times 9\left(B\right)$$$$In\left(A\right)and\left(B\right)thereisonlycase$$$$withequala+b+c+d=18$$$$a+c=9\Rightarrow a=9-c$$$$b+d=9\Rightarrow d=9-b=c-3$$$$b+c=12\Rightarrow b=12-c$$$$a+d=6$$$$\left(ii\right)and\left(iii\right)$$$$a(c+d)+b(c+d)=77$$$$(a+b)(c+d)=11\times 7$$$$a+b+c+d=18$$$$21-2c=11\Rightarrow c=5$$$$a=4$$$$d=2$$$$b=7$$$$a=4,b=7,c=5,d=2$$$$a+b+c+d=18$$

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