If 460 is divided by a natural number , quotient is 6 more than five times the divisor and remainder is 1. find the quotient and divisor.

Answer

To solve the problem step by step, we can follow the reasoning laid out in the video transcript:

### Step 1: Define the Variables
Let the divisor be \( x \). According to the problem, the quotient is \( 6 \) more than \( 5 \) times the divisor. Therefore, we can express the quotient as:
\[ \text{Quotient} = 5x + 6 \]

### Step 2: Write the Division Equation
According to the division algorithm, we have:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
Given that the dividend is \( 460 \) and the remainder is \( 1 \), we can write:
\[ 460 = x(5x + 6) + 1 \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ 460 = x(5x + 6) + 1 \]
Subtract \( 1 \) from both sides:
\[ 459 = x(5x + 6) \]
Expanding the right side gives:
\[ 459 = 5x^2 + 6x \]

### Step 4: Rearrange into Quadratic Form
Rearranging this equation to standard quadratic form gives:
\[ 5x^2 + 6x - 459 = 0 \]

### Step 5: Solve the Quadratic Equation
To solve the quadratic equation \( 5x^2 + 6x - 459 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 5 \), \( b = 6 \), and \( c = -459 \).

Calculating the discriminant:
\[ b^2 - 4ac = 6^2 - 4 \cdot 5 \cdot (-459) = 36 + 9180 = 9216 \]

Now, applying the quadratic formula:
\[ x = \frac{-6 \pm \sqrt{9216}}{2 \cdot 5} \]
Calculating \( \sqrt{9216} = 96 \):
\[ x = \frac{-6 \pm 96}{10} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{90}{10} = 9 \)
2. \( x = \frac{-102}{10} = -10.2 \) (not a natural number)

Thus, the only valid solution is:
\[ x = 9 \]

### Step 6: Find the Quotient
Now, substituting \( x = 9 \) back into the equation for the quotient:
\[ \text{Quotient} = 5(9) + 6 = 45 + 6 = 51 \]

### Final Answer
The divisor is \( 9 \) and the quotient is \( 51 \).