To determine how many 3-digit integers are divisible by 4, we'll follow a step-by-step process:
1. Identify the range of 3-digit integers:
Three-digit integers range from 100 to 999.
2. Find the first 3-digit number divisible by 4:
Starting from 100, we check the divisibility by 4. We find that 100 is divisible by 4. Therefore, the first 3-digit number divisible by 4 is 100.
3. Find the last 3-digit number divisible by 4:
Starting from 999, we check the divisibility by 4. The closest number below 999 that is divisible by 4 is 996. Hence, the last 3-digit number divisible by 4 is 996.
4. Count the numbers divisible by 4:
The number of terms in an arithmetic sequence can be determined using the formula for the number of terms in a sequence:
[tex]\[ \text{Number of terms} = \frac{\text{(last term - first term)}}{\text{common difference}} + 1 \][/tex]
Here, the first term (a) is 100, the last term (l) is 996, and the common difference (d) is 4.
Plugging in the values:
[tex]\[
\text{Number of terms} = \frac{(996 - 100)}{4} + 1 = \frac{896}{4} + 1 = 224 + 1 = 225
\][/tex]
Thus, there are 225 three-digit integers that are divisible by 4.
