First test case:
In this test case we can assume M as P₁² and N as Q₁¹. Their product will be S=P1².Q1¹, for minimum number of factors we can assume P1 equal to Q1 and in that case S will be P₁³ and the number of factors will be 1,P₁¹,P₁² and P₁³. Note that P₁ is a prime number. For maximum number of factors we will assume them to be different and factors in that case will be 1,P₁¹,Q₁¹,P₁²,P₁¹Q₁¹,P₁²Q₁¹.
Minimum and maximum number of factors are 4 and 6 respectively.

Second test case:

M=P₁²P₂² and N=Q₁¹Q₂³Their product S=M*N=P₁²P₂²Q₁¹Q₂³
For minimum number of factors we can assume P₁ and Q₁ equal and P₂ and Q₂ equal and S will be P₁³Q₁⁵and number of factors in this case are 24.
Number of factors=(3+1)*(5+1)=24.
Note that we can not assume P₁ equal to P₂ because that is a constraint in the probelm statement that P₁ and P₂ are different.
For maximum we can assume P₁,P₂,Q₁ and Q₂ different.