| Task | Efficient Approach | |------|--------------------| | Find all factor pairs of a number ≤ 100M | Loop from 1 to sqrt(N), check divisibility. O(√N) time. | | Count how many times a product appears | For given product P, count its divisors ≤ 10,000. | | Generate all unique products | Double loop with pruning: for i=1 to 10,000, for j=i to 10,000, compute i*j and add to set. | | Determine if a number is in the table | For any number X ≤ 100M, if X has a divisor ≤ 10,000, it appears. | | Visualize density | Use logarithmic binning: group products into log10 intervals and plot frequency. |

for listing all distinct products without storing 100M entries:

Introduction When we hear "multiplication chart," most of us picture the familiar 10x10 or 12x12 grid from elementary school—a handy tool for learning times tables. But what happens when we scale that concept to its logical extreme: a multiplication chart covering factors from 1 all the way to 10,000?