5 deleted 38 characters in body
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C# (Visual C# Interactive Compiler), 97 93 91 87 90 bytes

x=>{for(int i=0,jd=0,d=5;++i<24;d^=6j;++i<24;d=5)for(j=0;++j<24;j=0;++j<24;d^=6)if(i*j%2+(i*j+1)*d%x<1)Print((i,j));}

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C# (Visual C# Interactive Compiler), 97 93 91 87 bytes

x=>{for(int i=0,j,d=5;++i<24;d^=6)for(j=0;++j<24;)if(i*j%2+(i*j+1)*d%x<1)Print((i,j));}

Saved 6 bytes thanks to Kevin Cruijssen!

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C# (Visual C# Interactive Compiler), 97 93 91 90 bytes

x=>{for(int i=0,d=0,j;++i<24;d=5)for(j=0;++j<24;d^=6)if(i*j%2+(i*j+1)*d%x<1)Print((i,j));}

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4 deleted 50 characters in body
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C# (Visual C# Interactive Compiler), 9197 93 91 87 bytes

x=>{for(int i=0,j;++i<24;j,d=5;++i<24;d^=6)for(j=0;++j<24;)if(i*j%2+(i*j+1)*(5-(i+j)%2*2)%x<1*d%x<1)Print((i,j));}

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C# (Visual C# Interactive Compiler), 91 bytes

x=>{for(int i=0,j;++i<24;)for(j=0;++j<24;)if(i*j%2+(i*j+1)*(5-(i+j)%2*2)%x<1)Print((i,j));}

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C# (Visual C# Interactive Compiler), 97 93 91 87 bytes

x=>{for(int i=0,j,d=5;++i<24;d^=6)for(j=0;++j<24;)if(i*j%2+(i*j+1)*d%x<1)Print((i,j));}

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3 deleted 2 characters in body
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C# (Visual C# Interactive Compiler), 9391 bytes

x=>{for(int i=0,j;++i<24;)for(j=0;++j<24;)if(i*j%2+(i*j+1)*(5-(i+j)%2>0?3:5%2*2)%x<1)Print((i,j));}

Saved 46 bytes thanks to Kevin CruissenCruijssen!

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C# (Visual C# Interactive Compiler), 93 bytes

x=>{for(int i=0,j;++i<24;)for(j=0;++j<24;)if(i*j%2+(i*j+1)*((i+j)%2>0?3:5)%x<1)Print((i,j));}

Saved 4 bytes thanks to Kevin Cruissen

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C# (Visual C# Interactive Compiler), 91 bytes

x=>{for(int i=0,j;++i<24;)for(j=0;++j<24;)if(i*j%2+(i*j+1)*(5-(i+j)%2*2)%x<1)Print((i,j));}

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2 added 36 characters in body
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