Yes, it would seem that M always = N
a) If M is not a multiple of 5, then no factor of M can be a factor of M + 5, so the LCM of (M, M+5) = M*(M+5).
This is not divisible by 5, so N cannot be divisible by 5, so the LCM(N,N+5) = N*(N+5).
Therefore M*(M+5) = N*(N+5)
Therefore, M = N.
b) If M is a multiple of 5, the LCM of (M, M+5)
= 5*LCM(M/5,(M+5)/5)
= 5*LCM(M/5,M/5+1)
= 5*(M/5)*(M/5+1)
This is divisible by 5, so N is divisible by 5, so the LCM(N,N+5) = 5*(N/5)*(N/5+1)
Therefore 5*(M/5)*(M/5+1) = 5*(N/5)*(N/5+1)
Therefore M*(M+5) = N*(N+5)
Therefore, M = N.
There is nothing unique about 5. For any prime P, with M and N positive integers,
M = N if and only if LCM(M,M+P) = LCM(N,N+P)
Edited on October 31, 2023, 7:53 pm
Analytical Solution (spoiler)