A number X is converted to base 7 and becomes a four-digit number. Its leftmost digit is removed and placed at the right of the remaining three digits, resulting in a new number in base 7 which is twice the value of X. Find the decimal representations of all such numbers X.
The only one I could find was 480
To solve this problem, we need to find the decimal representations of all possible values of X. Let's break down the problem step by step.
Step 1: Convert the four-digit number from base 7 to decimal.
Let's use the following notation for the four-digit number in base 7: ABCD.
The decimal representation of this number can be found using the following formula:
Decimal = A * 7^3 + B * 7^2 + C * 7^1 + D * 7^0
Step 2: Remove the leftmost digit and place it at the right of the remaining three digits.
After moving the leftmost digit to the right, the resulting number in base 7 can be represented as: BCDA.
Step 3: Convert the new number from base 7 to decimal and check if it is twice the value of X.
Again, we can use the formula mentioned in Step 1 to convert BCDA to decimal.
To find the decimal representations of all such numbers X, we will follow these steps for all possible values of A, B, C, and D (which are digits from 0 to 6 in base 7).
Let's write a program or create a loop to iterate through all the possible values of A, B, C, and D and calculate the decimal representations for each case.
Here's a Python code snippet that implements this logic:
```python
def find_X_numbers():
for A in range(7):
for B in range(7):
for C in range(7):
for D in range(7):
decimal_X = A * 7**3 + B * 7**2 + C * 7**1 + D * 7**0
decimal_new = B * 7**3 + C * 7**2 + D * 7**1 + A * 7**0
if decimal_new == 2 * decimal_X:
print(decimal_X)
find_X_numbers()
```
This code will iterate through all possible values of A, B, C, and D (from 0 to 6 in base 7) and check if the resulting number is twice the decimal representation of X. It will print all the decimal representations of X that satisfy the conditions.
Note: This code may take some time to run since it needs to iterate through a large number of combinations.