In what number base is the addition 465 + 24 + 225 = 1050?
Answer Details
To solve this problem, we need to find out in what base the addition statement is true. Let's assume the base is "b".
Then, in the units column, we have:
- 5 + 4 = 10, so we write down 0 and carry-over 1.
- In the "b" column, we have: 6 + 2 + 2 + 1 = 11, so we write down 1 and carry-over 1.
- In the "b^2" column, we have: 4 + 2 + 5 + 1 = 12, so we write down 2 and carry-over 1.
- In the "b^3" column, we have: 4 + 2 + 2 = 8.
Putting these digits together, we get the number 8201 in base "b".
Now, we need to check if this number is equal to 1050 in base 10.
8201 in base "b" means:
8 x b^3 + 2 x b^2 + 0 x b + 1 x 1 = 1050
Rearranging the terms, we get:
8 b^3 + 2 b^2 + 1 = 1050
Subtracting 1 from both sides:
8 b^3 + 2 b^2 = 1049
Since b is a positive integer, we can see that b must be greater than 5. Trying b = 6, we get:
8 x 6^3 + 2 x 6^2 = 1048
This is not equal to 1049, so we need to try a larger base. Trying b = 7, we get:
8 x 7^3 + 2 x 7^2 = 1049
This is equal to 1049, so the base is 7. Therefore, the answer is seven.