If 314 base 10-256 base 7=340 base x, find x.
To solve this problem, we will convert both numbers from base 7 to base 10 and then subtract them.
Converting 314 base 10 to base 7:
314 ÷ 7 = 44 remainder 2
44 ÷ 7 = 6 remainder 2
6 ÷ 7 = 0 remainder 6
So, 314 base 10 is equal to 626 base 7.
Converting 256 base 7 to base 10:
(2 × 7^2) + (5 × 7^1) + (6 × 7^0) = 98 + 35 + 6 = 139
Now we subtract the base 10 numbers: 626 - 139 = 487.
Finally, we need to convert the difference (487 base 10) to base x:
487 ÷ x = q remainder r
Since r must be less than x, we know that r = 0.
487 = qx
487 = q(x)
q = 487
Since q is a number in base x, q must be less than x. However, the largest digit in base 7 is 6, which is smaller than x. Therefore, x must be greater than or equal to 7.
So, the possible values for x are 7, 8, 9, 10, and so on.
To solve this problem, we need to convert both numbers to the same base (base x) in order to subtract them.
First, let's convert 314 base 10 to base x.
To convert a number from base 10 to base x, we need to repeatedly divide the number by x until the quotient becomes zero.
Let's start dividing 314 by x:
314 divided by x = quotient 1 remainder (x - 1)
(x - 1) divided by x = quotient 0 remainder (x - 1)
Since the quotient is zero, we stop the division. The remainders in each step are the digits of the number in base x.
So, the number 314 base 10 is represented as (x - 1)(x - 1) in base x.
Next, let's convert 256 base 7 to base x.
256 base 7 = (2 x 7^2) + (5 x 7^1) + (6 x 7^0) = 2(49) + 5(7) + 6 = 98 + 35 + 6
Now, we have to subtract (x - 1)(x - 1) from 98 + 35 + 6.
(98 + 35 + 6) - [(x - 1)(x - 1)]
Let's simplify this equation:
98 + 35 + 6 = 139
(x - 1)(x - 1) = (x^2 - 2x + 1)
Now, we can rewrite the equation:
139 - (x^2 - 2x + 1) = 340
To solve for x, we need to rewrite the equation in quadratic form:
x^2 - 2x + 1 - 139 + 340 = 0
x^2 - 2x + 202 = 0
Now, we can solve this quadratic equation for x using factoring, completing the square, or using the quadratic formula.
By factoring:
(x - 14)(x - 14) = 0
So, x = 14 is the solution to the equation.
Therefore, x = 14.