Gọi ƯCLN(3n+2;7n+1)=dƯCLN(3n+2;7n+1)=d (d là ước nguyên tố)

Ta có: ⎧⎨⎩3n+2⋮d7n+1⋮d{3n+2⋮d7n+1⋮d

=>⎧⎨⎩7(3n+2)⋮d3(7n+1)⋮d=>{7(3n+2)⋮d3(7n+1)⋮d

=>⎧⎨⎩21n+14⋮d21n+3⋮d=>{21n+14⋮d21n+3⋮d

=>(21n+14)−(21n+3)⋮d=>(21n+14)−(21n+3)⋮d

=>11⋮d=>11⋮d

=>d∈{±1;±11}=>d∈{±1;±11}

Do d là nguyên tố nên d=±11d=±11

=>3n+2⋮11=>3n+2⋮11

=>3n+11−9⋮11=>3n+11−9⋮11

=>3n⋮̸11=>3n⋮̸11

Do ƯC(3;11)=1ƯC(3;11)=1

=>n⋮̸11=>n⋮̸11

=>=> Để p/s tối giản thì n=11k