Multiplication without figures I have taken the 12th problem of this pdf, do you know any way to resolve it without using brute force?
Simply I have to replace '*' of this multiplication below with correct digits, in order to have the factors corrected.
$$\begin{alignedat}{2}*&*4&&\,\times{}\\ &**&&={}\\\hline 44&*4&&  \end{alignedat} $$
Each '*' can be a different digit.

*

*Hints of bruteforceless method without factorising:


$\ \ \ \ \  \  abc\ \ $
$*\ \ \ \  \ de$
$=xyzt$
1- $t=ce=4e$, returning to rule of congruence $e$ is either $1$ or $6$ whilst $r(t)=2$ the retained
2- Since $x=ad+r(y)=4$ so either ${a,d}=2,r(y)=0$ or $a=1$ and $d={1,2,3}$ or vice versa with $r(y)$ is respectively {3,2,1}
3- Sine $ae+bd=4$ starting with case $e=1$
$a+bd+r(z)=4+r(y)$ if $(a,d)=(2,2),r(y)=0$ then $b=1$ and $r(z)=0$ so $z=dc+be=4*2+1*1=9$ is a valid solution
