All numbers divisible by 21 with remainder 1 can be written 21n+1 where n is an integer. All 3-digit numbers are less than 1000, so 21n+1<1000, n<999/21, n<47.6 so max n=47, meaning that there are 47 such numbers.
The LCM of 21 and 35 is 105, and all numbers divisible by both 21 and 35 with remainder 1 can be written 105m+1 where m is an integer. So 105m+1<1000, 105m<999, m<999/105, m<9.5 so max m=9. There are 9 numbers divisible by both 21 and 35 with remainder 1. So there are 47-9=38 numbers divisible by 21 remainder 1, which are divisible by 35 with remainder not equal to 1. 38 numbers are left in the list.