Find all n∈N such that √((9n-8)/(4n+5)) ∈ Q, HCF Linear Combination of the two integers is useful

Any linear combination of two integers will be divisible by the Greatest Common Divisor of the two integers. A slightly more difficult math problem than the one presented by the prolific gymnast mathematician and Virginian at Randolph College(Doctor Peter Sheldon a caring and kind physics educator). @andik70 questioned whether Bezout's identity/lemma was necessary regarding the linear combination property that states the Greatest Common Divisor is a divisor of ANY linear combination of the two integers.