Question

solve long division method 99999

Answer 1

(10x)² + 2*(10x)*y + y² where y∈{0..9} 

x²100 + 2*(10x)*y + y² 

√999`99 => √999 


(10u)² + 2*(10u)*v + v² where v∈{0..9} 

u²100 + 2*(10u)*v + v² 

√99`9 => u=3 

99`9 - u²100 = 99`9 - 900 = 99 =2*(10u)*v + v² =(2*10u+v)v = (2*10*3+v)v = (60+v)v and v∈{0..9} 
99 = (60+v)v 
Hence 
9`9 = (60+v)v => 9 div 6 =1 
Let v=1 
(60+1)*1=61 < 99 => 10u+v=31 

Hence 
√999`99 
(10x)² + 2*(10x)*y + y² where y∈{0..9} and x=31 
999`99 - (10x)² = 999`99 - 100*31² =999`99 - 96100=3899 

Hence 
3899 = 2*(10x)*y + y² = 2*(10*31)*y + y² = 620y + y² 
389`9 = 620y + y² => 389 div 62 =6 

Let y=6 
2*(10x)*y + y² = 2*(10*31)*6 + 6²= 620*6 + 36 =3756 < 3899 

Hence 
10x+y=10*31+6=316 

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