It’s not too tricky, but the key here (since it’s a math olympiad!) is to explain the solution step by step instead of just shouting out the answer. So, let’s dive in!

How Many 4-Digit Numbers Are Not Divisible by 998 and Have Even First and Last Digits?

Step 1. Let’s start with the basics. How many 4-digit numbers are there? Easy: from 1000 to 9999, we have 9,000 numbers in total.

Step 2. Now, the first digit has to be even. That means we’re looking at numbers starting with 2, 4, 6, or 8. Since there are four options for the first digit, this drops our count to 4,000 numbers.

Step 3. Next, the last digit has to be even, too. So, the valid last digits are 0, 2, 4, 6, or 8. Only half of our 4,000 numbers have even last digits, leaving us with 2,000 numbers.

Step 4. Here’s where 998 enters the picture. We need to remove all the numbers divisible by 998 from our list. But are they still there? Could we have removed them already? Nope! All numbers divisible by 998 are even, so they’re still in the mix.

The quick estimate suggests one per thousand numbers starting with 2, 4, 6, or 8. The calculation confirms it with the numbers 2994, 4990, 6986, and 8982. That’s 4 numbers we need to exclude.

Step 5. Finally, we subtract these 4 numbers from our total of 2,000. That leaves us with 1,996 numbers. There you have it!

Last modified on 2024-12-12