What is a 4-Digit Number Divisible by Both 5 and 9?

Understanding the concept of a 4-digit number divisible by both 5 and 9 can be quite intriguing. This article will explore the rules for divisibility by these numbers and demonstrate how to find such a number step by step.

Understanding Divisibility Rules

To determine whether a number is divisible by both 5 and 9, it is crucial to understand the specific rules:

Divisibility by 5

A number is divisible by 5 if its last digit is either 0 or 5. This means that if the number ends with a 0 or 5, it is divisible by 5.

Divisibility by 9

A more nuanced rule states that a number is divisible by 9 if the sum of its digits is divisible by 9. This means that if you add up all the digits of a number and the result is a multiple of 9, the number itself is divisible by 9.

Formulating a 4-Digit Number

Let's denote a 4-digit number as N 1000a 100b 10c d, where a, b, c, and d are its digits. The digit a must be between 1 and 9 to ensure the number remains a 4-digit number.

Step 1: Last Digit

To meet the divisibility rule for 5, the last digit (d) must be either 0 or 5. We will explore both possibilities.

Case 1: d 0

Let's consider an example where d 0.

If we take a 1, b 0, c 8, d 0: The number is 1080. The sum of the digits: 1 0 8 0 9, which is divisible by 9.

Thus, 1080 is a valid 4-digit number that is divisible by both 5 and 9.

Case 2: d 5

Now let's consider the case where d 5.

For example, if we take a 1, b 0, c 4, d 5:

The number is 1045. The sum of the digits: 1 0 4 5 10, which is not divisible by 9.

Another combination: if we take a 1, b 8, c 5, d 5:

The number is 1855. The sum of the digits: 1 8 5 5 19, which is not divisible by 9.

We continue this process to find valid combinations:

If we take a 1, b 2, c 3, d 5: The number is 1235. The sum of the digits: 1 2 3 5 11, which is not divisible by 9.

After checking various combinations, we confirm that 1080 is a valid solution.

Conclusion

Therefore, 1080 is a 4-digit number that is divisible by both 5 and 9.

Additional Solutions

There are other 4-digit numbers that meet this criteria:

1800 (180 × 10 1800; 1800 ÷ 180 10 / check) 9000 (90 × 100 9000; 9000 ÷ 90 100 / check) 1350 (135 × 10 1350; 1350 ÷ 10 135 / check) 4500 (45 × 100 4500; 4500 ÷ 45 100 / check)

These numbers are valid solutions as they meet the criteria of being divisible by both 5 and 9.