A 999-digit number starts with 9. Every 2 consecutive digits is divisible by 17 or 23. There are 2 possibilities for the last 3 digits. What is the sum of these 2 possibilities?
To find the sum of the two possibilities for the last three digits of the given number, let's break down the problem step by step.
Step 1: We know that the number starts with 9 and has 999 digits in total. So, we can write the number in the form: 9 _ _ _ _ _ ... _ _ _ (where the underscores represent the unknown digits).
Step 2: Since every two consecutive digits in the number are divisible by either 17 or 23, it means that the number formed by any two consecutive digits must be divisible by 17 or 23.
Step 3: Let's consider the first two digits after the initial 9. These two digits must form a number that is divisible by either 17 or 23.
Step 4: There are only a limited number of combinations of two digits that can be formed from the numbers 0 to 9, inclusive. We can find all possible pairs of two-digit numbers and check if each pair is divisible by either 17 or 23.
Step 4a: Create a list of all two-digit numbers from 10 to 99 (inclusive).
Step 4b: Iterate through each pair of two-digit numbers and check if each pair is divisible by either 17 or 23.
Step 4c: Remove any pairs that are not divisible by either 17 or 23 from the list.
Step 5: Next, let's find which combinations we can use for the remaining digits. Since the unknown digits can range from 0 to 9 (inclusive), there are 10 choices for each of the remaining 997 digits.
Step 6: Now that we have the list of valid combinations for the first two digits and 10 choices for each of the remaining 997 digits, we can calculate the number of possible numbers that satisfy the given conditions.
Step 6a: Multiply the number of combinations for the first two digits by the number of choices for each of the remaining 997 digits.
Step 6b: Multiply the result by 2 since there are two possibilities for the last three digits.
Step 7: Calculate the sum of the two possibilities for the last three digits by summing the product obtained in step 6b.
By following these steps, we can determine the sum of the two possibilities for the last three digits of the given number.