A 3-digit number is a multiple of 6 → divisible by 2 and divisible by 3 . → Last digit must be even ( 2,4,6 ). → Sum of digits must be divisible by 3 . Step 1: Choose last digit (must be even) Possible last digits =2,4,6⟶3 choices Step 2: After choosing the last digit, the remaining 5 digits include: Residues mod 3: - {1,4}⟶ remainder 1 - {2,5}⟶ remainder 2 - {3,6}⟶ remainder 0 No matter which even digit you remove, the remaining residues always allow 4 valid pairs whose divisible by 3 : - One pair from (1-group, 2-group): 2×2=4 - (0-group, 0-group) gives no pair - (1-group, 1-group) or (2-group, 2-group) invalid for 3-digit sum Therefore: 4 valid digit-pairs for the first two positions. Each pair can be arranged in 2 ways → total =8 numbers per last digit. Step 3: Multiply 3‌ (last digit choices) ‌×8=24‌. ‌