1. a door code is made up of six unique digits the last digit is 8.
2.oddd and even digits alternate(o is even)
3.the difference of the adjacent is always bigger than 1.
4.the first 2 digits (reads as one number)as well as he two middle digits(as one number) are multiples of the last two digits (as one number)
If the last digit is 8 the last two digits must be 18. A larger number ending in 8 would not have a two-digit multiple.
723618 could be the number. 367218 also works except for having the adjacent 6 and 7 differ by 1.
To find the door code, we need to follow the given conditions step by step:
1. The last digit is 8: Since the last digit is already given as 8, we have one digit of the code (the last digit).
2. Odd and even digits alternate, and 0 is even: Given that odd and even digits alternate and 0 is even, we can deduce that the digits in the code will follow the pattern EOEOEO... (where E represents even and O represents odd).
3. The difference between adjacent digits is always bigger than 1: This condition tells us that there cannot be consecutive digits in the code. So, for example, if one digit is 4, the next digit cannot be 3 or 5.
4. The first 2 digits (read as one number) and the two middle digits (read as one number) are multiples of the last two digits (read as one number): This condition indicates that the first two digits (considered as one number) and the middle two digits (considered as one number) must be divisible by the last two digits (considered as one number).
Let's proceed to find the door code by following these conditions:
Step 1: The last digit is 8. So far, we have "_ _ _ _ _ 8".
Step 2: Apply the alternating pattern of odd and even digits. Since the last digit is even, the next digit must be odd. We have two possibilities for the second-to-last digit: 1 or 3. If we choose 1, the pattern continues as EOEOE. If we choose 3, the pattern continues as EOEOEO. Let's consider both possibilities.
Case 1: If the pattern is EOEOE, the code would look like "_ _ _ _ 3 8".
- The first two digits (considered as one number) and the two middle digits (considered as one number) must be multiples of the last two digits (considered as one number). Since the last two digits are 3 and 8, we need to find two numbers that meet this condition. Let's consider some examples:
- If the first two digits are 16, and the two middle digits are 38, we have "16 38 3 8". In this case, both the first two digits and the two middle digits are multiples of the last two digits, so this is a possible code.
- If the first two digits are 24, and the two middle digits are 38, we have "24 38 3 8". In this case, the two middle digits are multiples of the last two digits, but the first two digits are not multiples of the last two digits. Therefore, this is not a valid code.
- So, if we choose the pattern as EOEOE and start with the last two digits as 3 and 8, a possible code could be "16 38 3 8".
Case 2: If the pattern is EOEOEO, the code would look like "_ _ _ _ 1 3 8".
- Again, the first two digits (considered as one number) and the two middle digits (considered as one number) must be multiples of the last two digits (considered as one number). Let's consider some examples:
- If the first two digits are 16, and the two middle digits are 38, we have "16 38 1 3 8". In this case, both the first two digits and the two middle digits are multiples of the last two digits, so this is a possible code.
- If the first two digits are 24, and the two middle digits are 38, we have "24 38 1 3 8". In this case, the two middle digits are multiples of the last two digits, but the first two digits are not multiples of the last two digits. Therefore, this is not a valid code.
- So, if we choose the pattern as EOEOEO and start with the last two digits as 1, 3, and 8, a possible code could be "16 38 1 3 8".
In summary, the possible door codes satisfying all the given conditions are "16 38 3 8" and "16 38 1 3 8".