A. What is the lowest base 12 positive integer that has exactly 10 (base 12) distinct positive factors?

B. Exactly 1,000 (base 12) distinct positive factors?

C. Exactly 1,000,000 (base 12) distinct positive factors?

For example, the distinct positive factors of 40 (base 12) are the base 12 numbers 1, 2, 3, 4, 6, 8, 10, 14, 20, and 40. Accordingly, 40 (base 12) has precisely A (base 12) distinct positive factors.

(In reply to re: thoughts on part B by Justin)

ran a quick program in Mathematica that checks for every possible way of factoring 1728 into factors greater than 2 and confirms that 54B,74B,200 is indeed the minimum

here is the code:

func[pows_]:=(

prd=1;

For[pi=1,pi„TLength[pows],++pi,

prd*=Prime[pi]^(pows[[pi]]-1);

];

Return[prd];

);

pow=3;

p2=2*pow;

p3=pow;

strt=Flatten[{Table[2,{i,1,p2}],Table[3,{i,1,p3}]}];

lst={strt};

biglst={lst};

minlst=strt;

min=func[minlst];

lng=Length[strt];

For[s=lng-1,s„d1,--s,

lst2={};

For[i=1,i„TLength[lst],++i,

facs=lst[[i]];

For[i1=1,i1„Tlng-1,++i1,

For[j1=i1+1,j1„Tlng,++j1,

p1=facs[[i1]];

p2=facs[[j1]];

p3=p1*p2;

lst3=Delete[facs,{{i1},{j1}}];

lst3=Sort[Append[lst3,p3],Greater];

If[!MemberQ[lst2,lst3],

AppendTo[lst2,lst3];

v=func[lst3];

If[v<min,

min=v;

minlst=lst3;

];

];

];];

];

--lng;

lst=lst2;

AppendTo[biglst,lst];

];

min

minlst

now for part C, I simply set pow=6, that will be the next post

Posted by Daniel on 2010-10-11 19:34:24