Find base n if 126n=86
To find the base, we need to solve for n. We can start by isolating n by dividing both sides of the equation by 126:
```
126n = 86
n = 86/126
```
We can simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 2:
```
n = (43/63)
```
We can see that the base must be a number that makes the fraction $\frac{43}{63}$ equal to a whole number. We can check different bases by expressing 43 and 63 in terms of that base:
```
43 = 4b + 3
63 = 6b + 3
```
We can see that any base that satisfies these equations would work. One possible solution is base 10, which gives:
```
43 = 4(10) + 3
63 = 6(10) + 3
```
So the answer is base 10.
To find the base, we can use the concept of place value.
In the given equation, 126n = 86, the left-hand side represents a number in base n, while the right-hand side represents a number in base 10.
Let's first convert the number 86 to base n. To do this, we'll look at the highest power of n that is less than or equal to 86. In this case, it is n^1.
n^1 = 86
Since n^1 is the same as n, we can write this equation as:
n = 86
So, the base n is 86.
To find the base (n) in the given equation 126n = 86, we can follow these steps:
Step 1: Set up the equation.
In a base-n number system, each digit's place value increases by n as you move from right to left. So, we can express the given equation as follows:
1 * n^2 + 2 * n^1 + 6 * n^0 = 8 * n^1 + 6 * n^0
Step 2: Simplify the equation.
Simplifying the equation gives us:
n^2 + 2n + 6 = 8n + 6
Step 3: Move all terms to one side.
To isolate the terms containing n, move all other terms to the other side of the equation:
n^2 + 2n - 8n + 6 - 6 = 0
Simplifying further, we have:
n^2 - 6n = 0
Step 4: Factor the equation.
To factor the equation, we look for two numbers whose product is 0 and whose sum is the coefficient of the linear term (-6n). In this case, n * (n - 6) = 0, so we have two potential solutions: n = 0 and n - 6 = 0.
Step 5: Solve for n.
Setting each factor equal to zero, we have:
n = 0
n - 6 = 0
n = 6
Therefore, the base (n) that satisfies the equation 126n = 86 is 6.