Let us assume that M is a nonnegative integer. Then we observe that the smallest value of M is zero, as writing M = 00....00( k zeros) we can trivially observe that it satisfies all the conditions of the problem.

Let us now assume that M is a positive integer.

Then, writing M as 10x+y, we must have:

3(10x+y)/2 = (10^n)*y+x

=> 2*(10^n)*y+2x=30x+y

=>  28x= y*(2*(10^n )-3)

=> 7*4*x = y*(2(10^n)-3) ....(#)

Now, 2*10^n-3 is obviously odd, and so:

 gcd(2*10^n-3, 4)=1

This is only possible if y is a multiple of 4.

Now, 7 must divide 2*10^n-3

Therefore, 2*10^n = 3 (mod 7) = 10 (mod 7)

=> 10^n = 5(mod 7)

Now, 10=3(mod 7) => 10^2 = 2(mod 7) => 10^3 = 6(mod 7) 

=> 10^4= 4 ( mod 7) => 10^5= 5(mod 7)

Accordingly,  from (#), we have:

(2*10^5-3)*y= 7*4*x

=> (199997/7)*y= 4*x

=> 28571*y= 4*x

Now, we recall that we have shown that y must be a multiple of 4. But since y is a single digit, we must have y=4,8

y=4 gives,  x= 28571

y=8 gives, x= 57142

Recalling that M =10*x+y, we observe that:

 M can assume the values 285714 or, 571428.

Consequently,  the required smallest number that satisfies al, the conditions  of the puzzle  under reference is 285714.

As a check, we can easily verify that: (3/2)*(285714) = 428571, which is perfectly in consonance with the given conditions,