LCM(2,5) = 10 so they will meet every 10 days.
Now, "in the next 30 days" is Day 2 to Day 31.
If they met on Day 1, then they will meet again on
Day 11, 21 and 31 (every 10 days).

Answer: They will meet 3 more times in the next 30 days.

Pictorial representation

  1  2  3  4  5  6  7  8  9 10
  K     K     K     K     K
  C              C

 11 12 13 14 15 16 17 18 19 20
  K     K     K     K     K
  C              C

 21 22 23 24 25 26 27 28 29 30
  K     K     K     K     K
  C              C

 31
  K
  C

They meet on the first day of the month, this means

day 2 for Kevin  = day 5 for cousin

So the other days they meet would be the highlighted days which coincides.

Multiples of 2 <=30  —>   2, 4, 6 , 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30

Multiples of 5 <= 30  —>   5, 10, 15, 20, 25, 30

Answer:  They will meet 3 more times in the next 30 days.

Nice.  Just wondering… is there anyway to solve this problem without drawing out the days?  I guess not.