Vinny wrote down all the single-digit base-b numbers and added them in base b, getting 34_b.
What is b?
the sum of the integers from 0 to b-1 is b(b-1)/2
so, b(b-1)/2 = 3b+4
b^2-b = 6b+8
b^2-7b-8 = 0
(b-8)(b+1) = 0
b=8 since b>1
check:
sum(0..7) = 28 = 3*8+3
To find the value of the base, b, we need to understand how addition works in different bases. In base b, we use digits ranging from 0 to (b - 1).
Vinny wrote down all the single-digit base-b numbers, which means he wrote the numbers from 0 to (b - 1). When these numbers are added in base b, the maximum sum would be the base b minus one repeated.
In this case, Vinny obtained a sum of 34_b. To understand this, we need to convert the sum to base 10. Let's do the conversion.
In base b, the digits represent powers of b. So, 34_b can be written as 3*b^1 + 4*b^0, which simplifies to 3b + 4.
Now, we can equate the sum 3b + 4 to the maximum sum of all single-digit base-b numbers, which is (b - 1) + (b - 1) + ... + (b - 1). Since there are b single-digit numbers, the maximum sum can be represented as b * (b - 1).
Equating the two equations:
3b + 4 = b * (b - 1)
Expanding the right side of the equation:
3b + 4 = b^2 - b
Rearranging and simplifying:
b^2 - 4b - b + 4 = 0
b^2 - 5b + 4 = 0
Now, we can solve this quadratic equation to find the value of b. We can factorize it as:
(b - 4)(b - 1) = 0
From this, we have two possible solutions: b = 4 and b = 1.
However, since we are looking for a single-digit base-b number, the base cannot be 1. Therefore, the value of b is 4.