Write it as a product
Number Theory
Level
4
Determine the number of unordered triplets of positive integers such that their product is
The answer is 139.
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1 solution
Mathematica
Length@Solve[a*b*c==1000000&&0<a<=b<=c,{a,b,c},Integers]139
here are the triplets
{{1,1,1000000},{1,2,500000},{1,4,250000},{1,5,200000},{1,8,125000},{1,10,100000},{1,16,62500},{1,20,50000},{1,25,40000},{1,32,31250},{1,40,25000},{1,50,20000},{1,64,15625},{1,80,12500},{1,100,10000},{1,125,8000},{1,160,6250},{1,200,5000},{1,250,4000},{1,320,3125},{1,400,2500},{1,500,2000},{1,625,1600},{1,800,1250},{1,1000,1000},{2,2,250000},{2,4,125000},{2,5,100000},{2,8,62500},{2,10,50000},{2,16,31250},{2,20,25000},{2,25,20000},{2,32,15625},{2,40,12500},{2,50,10000},{2,80,6250},{2,100,5000},{2,125,4000},{2,160,3125},{2,200,2500},{2,250,2000},{2,400,1250},{2,500,1000},{2,625,800},{4,4,62500},{4,5,50000},{4,8,31250},{4,10,25000},{4,16,15625},{4,20,12500},{4,25,10000},{4,40,6250},{4,50,5000},{4,80,3125},{4,100,2500},{4,125,2000},{4,200,1250},{4,250,1000},{4,400,625},{4,500,500},{5,5,40000},{5,8,25000},{5,10,20000},{5,16,12500},{5,20,10000},{5,25,8000},{5,32,6250},{5,40,5000},{5,50,4000},{5,64,3125},{5,80,2500},{5,100,2000},{5,125,1600},{5,160,1250},{5,200,1000},{5,250,800},{5,320,625},{5,400,500},{8,8,15625},{8,10,12500},{8,20,6250},{8,25,5000},{8,40,3125},{8,50,2500},{8,100,1250},{8,125,1000},{8,200,625},{8,250,500},{10,10,10000},{10,16,6250},{10,20,5000},{10,25,4000},{10,32,3125},{10,40,2500},{10,50,2000},{10,80,1250},{10,100,1000},{10,125,800},{10,160,625},{10,200,500},{10,250,400},{16,20,3125},{16,25,2500},{16,50,1250},{16,100,625},{16,125,500},{16,250,250},{20,20,2500},{20,25,2000},{20,40,1250},{20,50,1000},{20,80,625},{20,100,500},{20,125,400},{20,200,250},{25,25,1600},{25,32,1250},{25,40,1000},{25,50,800},{25,64,625},{25,80,500},{25,100,400},{25,125,320},{25,160,250},{25,200,200},{32,50,625},{32,125,250},{40,40,625},{40,50,500},{40,100,250},{40,125,200},{50,50,400},{50,80,250},{50,100,200},{50,125,160},{64,125,125},{80,100,125},{100,100,100}}