Question

Among the 4-digit numbers formed using the digits \( 0, 1, 2, 3, 4 \) when repetition of digits is allowed, the number of numbers which are divisible by 4 is:

• A. \( 140 \)
• B. \( 160 \)
• C. \( 180 \)
• D. \( 200 \)

Hint:

For divisibility rules, focus on the specific digit placement that determines divisibility (e.g., last two digits for divisibility by 4).

Answer 1

The Correct Option is B

To determine the number of 4
-digit numbers formed using the digits \( 0, 1, 2, 3, 4 \) (with repetition allowed) that are divisible by 4, we proceed as follows: Step 1: Total number of 4
-digit numbers A 4
-digit number cannot start with \( 0 \). Therefore:
- The first digit (thousands place) can be \( 1, 2, 3, \) or \( 4 \) (4 choices).
- The second, third, and fourth digits (hundreds, tens, and units places) can be \( 0, 1, 2, 3, \) or \( 4 \) (5 choices each). Thus, the total number of 4
-digit numbers is: \[ 4 \times 5 \times 5 \times 5 = 500. \] Step 2: Divisibility by 4 A number is divisible by 4 if the last two digits form a number that is divisible by 4. Therefore, we need to count the number of valid 4
-digit numbers where the last two digits form a number divisible by 4. Step 3: Count valid last two digits The last two digits (tens and units places) must form a number divisible by 4. Using the digits \( 0, 1, 2, 3, 4 \), the possible valid pairs for the last two digits are: \[ 00, 04, 12, 16, 20, 24, 32, 36, 40, 44. \] However, since the digits available are \( 0, 1, 2, 3, 4 \), the valid pairs are: \[ 00, 04, 12, 20, 24, 32, 40, 44. \] There are 8 valid pairs for the last two digits. Step 4: Count valid 4
-digit numbers For each valid pair of last two digits:
- The first digit (thousands place) can be \( 1, 2, 3, \) or \( 4 \) (4 choices).
- The second digit (hundreds place) can be \( 0, 1, 2, 3, \) or \( 4 \) (5 choices). Thus, for each valid pair of last two digits, there are: \[ 4 \times 5 = 20 \] valid 4
-digit numbers. Since there are 8 valid pairs, the total number of 4
-digit numbers divisible by 4 is: \[ 8 \times 20 = 160. \] Final Answer: \[ \boxed{160} \]