## Eight Knights

Eight dashing knights are in love with [the] kingdom’s princess. Starting on January $1$, the first knight visits the princess every day. The second night visits the princess on January $2$ and every second day thereafter. The third knight visits the princess on January $3$ and every third day thereafter. The pattern continues for each of the $8$ knights. What is the total number of knight visits up to and including the first day on which all $8$ knights visit the princess?
Source: mathcontest.olemiss.edu 9/3/2012

SOLUTION

Let’s simplify the problem by assuming there are only $4$ knights. We make a little spreadsheet to detect any pattern.

The first time all $4$ knights visit the princess is on day $12$ which is the least common multiple (LCM) of $1,2,3,4$.
During those $12$ days, the number of visits by each knight is as follows
Knight $1\!: 12$ times $\left ( 12\div 1=12\right )$
Knight $2\!: 6$ times $\left ( 12\div 2=6\right )$
Knight $3\!: 4$ times $\left ( 12\div 3=4\right )$
Knight $4\!: 3$ times $\left ( 12\div 4=3\right )$
Total number of visits
$12+6+4+3=25$

Applying the pattern to 8 knights
LCM of $1,2,3,4,5,6,7,8 = 840$
During those $840$ days, the number of visits by each knight is as follows
Knight $1\!: 840$
Knight $2\!: 420$
Knight $3\!: 280$
Knight $4\!: 210$
Knight $5\!: 168$
Knight $6\!: 140$
Knight $7\!: 120$
Knight $8\!: 105$
Total number of visits
$840+420+280+210+168+140+120+105=2283$

Answer: $2283$.