I came up with a poor 257, but it scrolls realistically... Oh well.

$d=10;$c;for(1..$d){$c=$c.int(rand(10));}print"$c\n";$g;$w='';$m=280;$
+t=$m;$|=1;do{$g='';for(1..$d){$g=$g.int(rand(10));}print"$w$g\r";if(s
+ubstr($g,0,1)eq substr($c,0,1)){if($t>0){$t--;}else{$w=$w.substr($c,0
+,1);substr($c,0,1)='';$d=$d- 1;$t=$m;}}}until($d==0)
[download]

Update: Got some tips from Albannach and got down to 204, but I'm spending characters for nice appearance:

$d=shift;for(1..$d){$c.=int(rand(10))}$t=$m=320;$|=1;while($d){$g='';f
+or(1..$d){$g.=int(rand(10))}print"$w$g\r";if(substr($g,0,1)eq substr(
+$c,0,1)){if($t){$t--;}else{$w.=substr($c,0,1,'');$d--;$t=$m;}}}
[download]

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