For the lhs to be defined, the denominator must be nonzero.

Accordingly, each of M and N must be nonzero.

Now, lhs = (10M+N)/(M*N) = (10+N/M)/N

and N < rhs <N+1

Accordingly, N(N+1) > 10 +N/M >10, since N and M are both non zero.

This is possible if N >=3 ........(i)

Again, N^2<10+N/M<=10+N

or, N^2-N <= 10

Now, for N=4, we have:4^2-4=12>10. This is a contradiction.

Therefore, N<=3........(ii)

Combining (i) and (ii), we must have N=3.

Accordingly, (10M+3)/(3M) =3.M3333.....

=> 10/3 +1/M = 3 + M/10 + 1 /30

=> 1/3 +1/M =M/10 +1/30

Multiplying both sides by 30*M, we have:

10*M+30=3 M^2+M

=> 3 (M^2-3M-10)=0

=>3(M+2)(M-5)=0

Hence, M=5, discarding the negative value which is inadmissible..

Consequently, (M,N) =(5,3), constitutes the only valid solution to the given problem.

(As a check,

lhs = (10*5+3)/(5*3) = 53/15 = 3.5333333........)

*Edited on ***December 14, 2021, 1:46 am**

*Edited on ***December 14, 2021, 2:19 am**