Suppose 2^m + 3ⁿ ⋮ 23

=> \(8^n\left(2^m+3^n\right)\text{⋮}23\)

\(\Rightarrow2^{m+3n}+24^n\text{⋮}23\)

Because \(24\equiv1\left(mod23\right)\Rightarrow24^n\equiv1\left(mod23\right)\)

\(\Rightarrow2^{m+3n}+24^n\equiv2^{m+3n}+1\left(mod23\right)\)

We must prove that 2^m+3n + 1 doesn't divisible by 23

Simple we can have a example: 2¹¹ \(⋮̸23\)

So \(2^m+3^n⋮23̸\)

Suppose 2m + 3n ⋮ 23

=> 8n(2m+3n)⋮23

⇒2m+3n+24n⋮23

Because 24≡1(mod23)⇒24n≡1(mod23)

⇒2m+3n+24n≡2m+3n+1(mod23)

We must prove that 2m+3n + 1 doesn't divisible by 23

Simple we can have a example: 211 ⋮̸ 23

So 2m+3n⋮23̸