Let the 4 parts be a, b, c, d. We can write: a+2=b-2=2c=d/2 and a+b+c+d=45, so b+c+d=45-a.
We can now write b, c, d in terms of a: b=a+4; 2c=a+2; d=2a+4; double the equations for b and d: 2b=2a+8, 2d=4a+8.
Now add the equations together:
2b=2a+8+
2c=a+2+
2d=4a+8
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2(b+c+d)=7a+18; 2(45-a)=7a+18; 90-18=9a, so a=72/9=8.
b=a+4=12; c=(1/2)(a+2)=5; d=2a+4=20. Therefore, the subdivisions of 45 are: 8, 12, 5 and 20, and the common number is 10 (8+2, 12-2, 2*5, 20/2).
(Now, where's that pot of gold?!)