N leaves a remainder of 4 when divided by 33, what are the possible remainders…

Short answer: There are 5 possible remainders: 4, 37, 15, 48, 26.

Key idea: Numbers congruent to 4 modulo 33 have the form 4 + 33k. When taken modulo 55, adding 33 cycles through a subgroup of size 55/gcd(33,55).

• Compute gcd(33, 55) = 11, so the number of distinct residues modulo 55 is 55/11 = 5.

• Generate 4 + 33k for k = 0,1,2,3,4: 4, 37, 70, 103, 136.

• Reduce these modulo 55 to get the distinct remainders 4, 37, 15, 48, 26.

Therefore the number of possible remainders when N is divided by 55 is 5. The correct choice is the one with the text "5".