Exploring 4-Digit Numbers Divisible by 25 Using Digits 0, 1, 2, 3, 5, 7

Are you curious about how many 4-digit numbers you can create using the digits 0, 1, 2, 3, 5, and 7 that are divisible by 25? If so, this article is for you! In this detailed guide, we will break down the process of forming 4-digit numbers divisible by 25 using the given digits and explore the combinatorial aspects involved.

Understanding Divisibility by 25

A number is divisible by 25 if its last two digits form a number that is divisible by 25. This means that the possible pairs of digits that can form such a number are:

00 25 50 75

However, since we are forming 4-digit numbers, the pair '00' cannot be used, as it would not result in a valid 4-digit number.

Case Analysis

Let's break down the problem into three cases based on the last two digits.

Case 1: Last two digits are 25

The first two digits can be any of 1, 2, 3, 5, or 7, with the first digit not being 0.

First digit: 5 choices Second digit: 6 choices (including 0)

Thus, the number of combinations for this case is:

5 × 6 30

Case 2: Last two digits are 50

The first digit can be any of 1, 2, 3, 5, or 7, again not being 0. The second digit can be any of 0, 1, 2, 3, 5, or 7.

First digit: 5 choices Second digit: 6 choices

The number of combinations for this case is:

5 × 6 30

Case 3: Last two digits are 75

The first digit can be any of 1, 2, 3, 5, or 7. The second digit can be any of 0, 1, 2, 3, 5, or 7.

First digit: 5 choices Second digit: 6 choices

The number of combinations for this case is:

5 × 6 30

Total Combinations

To find the total number of 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 5, and 7 and are divisible by 25, we sum the combinations from all three cases:

30 30 30 90

Therefore, the total number of 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 5, and 7 that are divisible by 25 is 90.

Additional Examples

To provide further clarity, let's list some of the 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 5, and 7 and are divisible by 25:

1025 1075 1250 1275 1325 1375 2075 2175 3025 3075 3125 3175 3275

Conclusion

By understanding the properties of numbers divisible by 25 and applying combinatorial methods, we have determined that there are 90 4-digit numbers that can be formed using the digits 0, 1, 2, 3, 5, and 7 and are divisible by 25. This article aims to provide a clear and detailed exploration of the topic, helping you grasp the concept effectively.

Keywords: 4-digit numbers, divisible by 25, combinatorics