To find the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5, which are divisible by 4, we need to follow a systematic approach. This question falls under the Quant category and requires a clear understanding of divisibility rules and counting techniques.


A number is divisible by 4 if the last two digits of the number are divisible by 4. Therefore, we need to focus on the last two digits of the 5-digit numbers we are forming.


To solve this problem, we can use the counting principle or multiplication principle. The multiplication principle states that if there are 'n' ways to perform one task and 'm' ways to perform another task, then there are n x m ways to perform both tasks together.


To form a 5-digit number, we need to consider the following:

1. Thousands place: We have 5 options (1, 2, 3, 4, 5) for the thousands place.
2. Hundreds, tens, and units places: We have 5 options (1, 2, 3, 4, 5) for each of these places since repetition is allowed.


Using the multiplication principle, we can calculate the total number of possibilities:

Number of possibilities = Number of options for thousands place x Number of options for hundreds place x Number of options for tens place x Number of options for units place

Number of possibilities = 5 x 5 x 5 x 5

Number of possibilities = 625


Out of the 625 possibilities, not all of them will be divisible by 4. We need to find the number of 5-digit numbers that have their last two digits divisible by 4.


To determine the possible last two digits, we need to consider the following:

1. The tens place can be filled with any of the digits 1, 2, 3, 4, and 5 (5 options).
2. The units place can be filled with any of the digits 1, 2, 3, 4, and 5 (5 options).


Using the multiplication principle, we can calculate the total number of possibilities for the last two digits:

Number of possibilities for last two digits = Number of options for tens place x Number of options for units place

Number of possibilities for last two digits = 5 x 5

Number of possibilities for last two digits = 25


To find the number of 5-digit numbers that are divisible by 4, we need to multiply the number of possibilities for the last two digits by the number of possibilities for the remaining three digits:

Number of 5-digit numbers divisible by 4 = Number of possibilities for last two digits x Number of possibilities for the remaining three digits

Number of 5-digit numbers divisible by 4 = 25 x 625

Number of 5-digit numbers divisible by 4 = 15625