Not sure if there is a shortest way to do this.
I just substituted values from 1 to 4 for n in both the equations 3n+2 and 8n+3.
When n=1, 3n+2= 5, 8n+3= 11 - No common divisor
When n=2, 3n+2= 8, 8n+3= 19 - No common divisor
When n=3, 3n+2= 11, 8n+3= 27 - No common divisor
When n=4, 3n+2=14 , 8n+3= 35 - divisible by 7
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