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heureka

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 #2
avatar+26396 
+3

Determine the smallest non-negative integer a that satisfies the congruences:

a2(mod3)a4(mod5)a1(mod7)a8(mod9)

 

1. join
a2(mod3)a8(mod9)

a=2+3na=8+9m2+3n=8+9m3n=6+9m|:3n=2+3ma=2+3n|n=2+3ma=2+3(2+3m)a=2+6+9ma=8+9ma8(mod9)a2(mod3)a8(mod9)}a8(mod9)

 

2. join
a4(mod5)a1(mod7)

a=4+5ra=1+7s4+5r=1+7s5r=7s3r=7s35r=5s+2s35r=s+2s35=tr=s+tt=2s355t=2s32s=5t+3s=5t+32s=4t+t+2+12s=2t+1+t+12=us=2t+1+uu=t+122u=t+1t=2u1s=2(2u1)+1+us=5u1r=5u1+2u1r=7u2a=4+5ra=4+5(7u2)a=6+35ua6(mod35)a4(mod5)a1(mod7)}a6(mod35)

 

3. join
a8(mod9)a6(mod35)

a=8+9va=6+35w8+9v=6+35w9v=35w14v=35w149v=27w+8w959v=3w1+8w59=xv=3w1+xx=8w599x=8w58w=9x+5w=9x+58w=8x+x+58w=x+x+58=yw=x+yy=x+588y=x+5x=8y5w=8y5+yw=9y5v=3(9y5)1+8y5v=35y21a=8+9va=8+9(35y21)a=181+315ya181(mod315)a181+315(mod315)a134(mod315)a8(mod9)a6(mod35)}a134(mod315)

 

The smallest non-negative integer a is 134

 

laugh

04.07.2021
 #1
avatar+26396 
+2

The interior angles of a convex polygon are in an arithmetic progression.
If the smallest angle is 100 degrees and common difference is  10 degrees ,
then find the number of sides.

 

Let the sides of the polygon n 


Arithmetic progression:sum=100+110+120+130++(100+(n1)10)sum=(100+(100+(n1)10)2)nsum=(100+100+10n102)nsum=(190+10n2)nsum=(95+5n)n

 

Interior angles of a convex polygon: sum=(n2)180

 

(95+5n)n=(n2)18095n+5n2=180n21805n285n+360=0|:5n217n+72=0n=17±1724722n=17±2892882n=17±12n=9orn=8

 

laugh

27.06.2021