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Question Number 188196 by cortano12 last updated on 26/Feb/23

$$\left(1\right)\mathrm{solve}\mathrm{Diopthantine}\mathrm{equation}$$$$754\mathrm{x}+221\mathrm{y}=13$$$$\left(2\right)\mathrm{find}\mathrm{the}\mathrm{number}\mathrm{abcd}$$$$\mathrm{such}\mathrm{that}4\times \left(\mathrm{abcd}\right)=\mathrm{dcba}$$

Commented by ARUNG_Brandon_MBU last updated on 26/Feb/23

#include <iostream> using namespace std; int main(void) { for (int i=1000; i<2500; i++) { for(int j=1001; j<10000;j++) { if (4*i == j) if (i%10 == j/1000) if (i%100/10 == j%1000/100) if (i%1000/100 == j%100/10) if (i/1000 == j%10) cout << "4*" << i <<"=" << j; } } return 0; } // output: 4*2178=8712

Commented by ARUNG_Brandon_MBU last updated on 26/Feb/23

$$a=2,b=1,c=7,d=8$$

Answered by ARUNG_Brandon_MBU last updated on 26/Feb/23

$$754x+221\mathrm{y}=13$$$$754=3\left(221\right)+91$$$$221=2\left(91\right)+39$$$$91=2\left(39\right)+13$$$$$$$$\Rightarrow 13=91-2\left(39\right)$$$$=91-2[221-2\left(91\right)]$$$$=-2\left(221\right)+5\left(91\right)$$$$=-2\left(221\right)+5[754-3\left(221\right)]$$$$=5\left(754\right)-17\left(221\right)$$$$\Rightarrow 754\left(5\right)+221(-17)=13$$$$\Rightarrow x=5,\mathrm{y}=-17$$

Answered by mr W last updated on 26/Feb/23

$$\left(2\right)$$$$4\times \left(abcd\right)=\left(dcba\right)$$$$abcd$$$$\underset{―}{\times 4}$$$$dcba$$$$a=1\left(rejected\right)or2\Rightarrow a=2✓$$$$4\times d=12\Rightarrow d=3\left(rejected\right)$$$$4\times d=32\Rightarrow d=8✓$$$$2bc8$$$$\underset{―}{\times 4}$$$$8cb2$$$$(2000+100b+10c+8)\times 4=8000+100c+10b+2$$$$13b+1=2c$$$$\Rightarrow b=1,c=7$$$$sothenumberis2178with$$$$2178\times 4=8712$$

Answered by manxsol last updated on 27/Feb/23

$$\overline{)\begin{array}{ccccccccc}d& c& b& a& & a& b& c& d\\ 4& 4& 1& 2& & 6& 9& 4& 4\\ 5& 5& 1& 1& & 4& 7& 6& 6\\ 6& 6& 1& 1& & 4& 7& 6& 6\\ 7& 7& 1& 1& & 8& 1& 7& 7\\ 8& 8& 2& 2& & 2& 5& 8& 8\\ 9& 9& 2& 2& & 2& 5& 8& 8\\ 9& 9& 2& 3& & 6& 9& 9& 9\\ & & & & & & & & \\ 8& 7& 1& 2& & 2& 1& 7& 8\end{array}}$$

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