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Part of 2000 Greece JBMO TST

Problems(1)

3n+5/2n+3 irreducible,( \overline{xy}+12)(\overline{xy}-3) \in N => x+y=9

Source: Greece JBMO TST 2000 p1

6/17/2019
a) Prove that the fraction 3n+52n+3\frac{3n+5}{2n+3} is irreducible for every n∈Nn \in N b) Let x,yx,y be digits of decimal representation system with x>0x>0, and xy‾+12xy‾−3∈N\frac{\overline{xy}+12}{\overline{xy}-3}\in N, prove that x+y=9x+y=9. Is the converse true?
Irreduciblenumber theoryDigits