The number of 4-digit numbers that can be formed using the digits 1, 2, 3, 5, 8 , if the repetition of digits is allowed and the number is divisible by 6, is _ _ _ _

Step-by-step solution

For a number to be divisible by 6, it must be even and the sum of its digits must be divisible by 3. Let the 4-digit number be d₁ d₂ d₃ d₄ . Since it must be even, the last digit d₄ must be even. The even digits in the given set 1, 2, 3, 5, 8 are 2 and 8 . Thus, d₄ 2, 8 , giving 2 choices for the last digit. Notice that both 2 and 8 are congruent to 2 3 . For the sum of the digits to be divisible by 3: d₁ + d₂ + d₃ + d₄ 0 3 d₁ + d₂ + d₃ + 2 0 3 d₁ + d₂ + d₃ 1 3 . The available digits modulo 3 are: 0 3 : 3 (1 digit)