If 2257 is the result of
subtracting 4577 from 7056 in
base n, find n
A. 8
B. 9
C. 10
D. 11
To find the base, we need to calculate the difference between 7056 and 4577 and express it in base n.
7056 - 4577 = 2479
Now we need to convert 2479 into base n.
The largest power of n that is less than or equal to 2479 is 3.
n^3 = 1000 is less than 2479.
n^2 = 100 is less than 2479.
n^1 = 10 is less than 2479.
So, we can express 2479 in base n as 200n^3 + 400n^2 + 700n^1 + 900n^0.
This means n = 9.
Therefore, the answer is B. 9.
To find the base (n), we can start by subtracting 4577 from 7056 in base 10:
7056 - 4577 = 2479
Now let's convert this result (2479) to base n. We'll divide the number by increasing powers of n until we reach a quotient less than n:
2479 ÷ n^3 = 2
2479 - 2n^3 = 479
479 ÷ n^2 = 4
479 - 4n^2 = 79
79 ÷ n^1 = 7
79 - 7n^1 = 0
Since the next quotient is 0, we can stop here and set up the following equation:
2n^3 + 4n^2 + 7n^1 = 0
Now we can test each choice for the value of n (from the given options: A, B, C, and D) by substituting it into the equation:
For option A (n = 8):
2(8)^3 + 4(8)^2 + 7(8)^1 = 0
1024 + 256 + 56 ≠ 0
For option B (n = 9):
2(9)^3 + 4(9)^2 + 7(9)^1 = 0
1458 + 648 + 63 ≠ 0
For option C (n = 10):
2(10)^3 + 4(10)^2 + 7(10)^1 = 0
2000 + 400 + 70 ≠ 0
For option D (n = 11):
2(11)^3 + 4(11)^2 + 7(11)^1 = 0
2662 + 484 + 77 = 3223
Since only option D (n = 11) gives us a sum of 0, the answer is:
D. 11