When a number x is divided by a divisor it is seen that the divisor = 4 times the quotient = double the remainder. If the remainder is 80 then the value of x is
6480
Understanding the Division Problem and Finding the Number
The problem asks us to find the value of a number, let's call it x, based on the relationships between the divisor, quotient, and remainder when x is divided by a divisor. We are given specific conditions that link these three components of the division process.
Given Information and Relationships
We are provided with the following information:
- A number
xis divided by a divisor. - The divisor is equal to 4 times the quotient. Mathematically, this can be written as: $$ \text{Divisor} = 4 \times \text{Quotient} $$
- The divisor is also equal to double the remainder. Mathematically, this can be written as: $$ \text{Divisor} = 2 \times \text{Remainder} $$
- The remainder is given as 80. Mathematically, this is: $$ \text{Remainder} = 80 $$
Step-by-Step Calculation to Find the Number x
Step 1: Find the Divisor
We know that the divisor is double the remainder, and the remainder is 80. Using the relationship Divisor = 2 × Remainder, we can find the divisor:
So, the divisor is 160.
Step 2: Find the Quotient
We also know that the divisor is 4 times the quotient. We have just calculated the divisor as 160. Using the relationship Divisor = 4 × Quotient, we can find the quotient:
To find the Quotient, we divide 160 by 4:
$$ \text{Quotient} = \frac{160}{4} $$ $$ \text{Quotient} = 40 $$So, the quotient is 40.
Step 3: Find the Number x using the Division Algorithm
The fundamental relationship in division, known as the Division Algorithm or Division Lemma, states that for any integer x (dividend) and a positive integer divisor, there exist unique integers quotient and remainder such that:
We have found the following values:
- Divisor = 160
- Quotient = 40
- Remainder = 80
Now, we can substitute these values into the Division Algorithm formula to find the number x:
First, calculate the product of the divisor and the quotient:
$$ 160 \times 40 = 6400 $$Next, add the remainder to this product:
$$ x = 6400 + 80 $$ $$ x = 6480 $$Conclusion
Based on the given conditions and calculations, the value of the number x is 6480.
| Component | Value |
|---|---|
| Remainder | 80 |
| Divisor | 160 |
| Quotient | 40 |
| Number (x) | 6480 |
Revision Table: Key Concepts in Division
| Term | Definition | Relationship |
|---|---|---|
| Dividend (x) | The number being divided. | \(x = \text{Divisor} \times \text{Quotient} + \text{Remainder}\) |
| Divisor | The number by which the dividend is divided. | Given relationships (e.g., \(4 \times \text{Quotient}\), \(2 \times \text{Remainder}\)) |
| Quotient | The result of the division (how many times the divisor fits into the dividend). | Found using given relationships involving Divisor. |
| Remainder | The amount left over after division. | Must be less than the Divisor (\(0 \le \text{Remainder} < \text{Divisor}\)) |
Additional Information: Division Algorithm Explained
The Division Algorithm is a foundational theorem in number theory. It precisely defines the relationship between a dividend, a divisor, a quotient, and a remainder. For any integer dividend 'a' and any positive integer divisor 'b', there are unique integers 'q' (quotient) and 'r' (remainder) such that:
$$ a = bq + r $$where the remainder 'r' satisfies the condition 0 ≤ r < b. This means the remainder is always non-negative and strictly less than the absolute value of the divisor. In our specific problem, the divisor is positive, so 0 ≤ Remainder < Divisor.
In this question:
- Dividend (a) is the number
xwe are looking for. - Divisor (b) is 160.
- Quotient (q) is 40.
- Remainder (r) is 80.
And we verified that $80 < 160$, satisfying the condition for the remainder.
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