Find the number of non-negative integral solutions. [x/19]=[x/21] , here [] denotes greatest integer function. How do we solve this ?

1 Answer
Dean R.
Jun 29, 2018

#19+17+15+13+11+9+7+5+3+1=100#

Explanation:

We have 19 matches from #x=0# to #x=18#. Then #x=19# and #20# fail so next cycle we get 17 matches, through #x=37#. Then four fails, 38, 39, 40, 41, then 15 matches. Each time we get two fewer matches, so the total is the sum of the odd numbers from one to 19. To use Gauss's trick,

#19+17+15+13+11+9+7+5+3+1#

#=19+1 + 17+3 + 15+5 + 13 + 7 + 11+9#

#=20+20+20+20+20#

#=100#

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