Problem:
Let K be the product of all factors (b−a) (not necessarily distinct) where a and b are integers satisfying 1≤a<b≤20. Find the greatest positive integer n such that 2n divides K.
Solution:
The product K contains nineteen 1's (2−1,3−2,4−3,…,20−19), eighteen 2's (3−1,4−2,5−3,…,20−18), and so forth. Thus K= 119⋅218⋅317⋅416⋯191. The power of 2 in this product is 218⋅416⋅214. 812⋅210⋅48⋅26⋅164⋅22. The number of factors of 2 is therefore 1⋅18+2⋅16+1⋅14+3⋅12+1⋅10+2⋅8+1⋅6+4⋅4+1⋅2=150.
The problems on this page are the property of the MAA's American Mathematics Competitions