the person removing them 5 at a time has 3 left follows the total number of eggs is 5k + 3, where k is integer

the person removing them 3 at a time has 2 left follows the total number of eggs is 3n + 2, where n is integer

the person removing them 2 at a time has 1 left follows the total number of eggs is 2m + 1, where m is integer

the total number of eggs is an odd number

we will try all the k's for which 5k + 3 is odd

k = 1 we have 5*1 + 3 = 8 is even so we stop here and go onto the next value for k

k = 2 we have 5*2 + 3 = 13 now we have

3n + 2 = 13 => n = 11/3 is not an integer

k = 3 we have 5*3 + 3 = 18 is even so we stop here and go onto the next value for k

k = 4 we have 5*4 + 3 = 23

3n + 2 = 23 => n = (23-2)/3 that is n = 7

2m + 1 = 23 => m = (23-1)/2 that is m = 11

the fewest number of eggs possible in the basket is 23.