Find the number by which 546 should be divided to get 7 as quotient and 7 as remainder
The Correct Answer and Explanation is:
To find the number by which 546 should be divided to get a quotient of 7 and a remainder of 7, use the basic division algorithm:
Dividend = (Divisor × Quotient) + Remainder
Let the unknown number be the divisor.
Given:
- Dividend = 546
- Quotient = 7
- Remainder = 7
Substitute the known values into the formula: 546=(Divisor×7)+7546 = (Divisor × 7) + 7546=(Divisor×7)+7
Subtract 7 from both sides: 546−7=Divisor×7⇒539=Divisor×7546 – 7 = Divisor × 7 \Rightarrow 539 = Divisor × 7546−7=Divisor×7⇒539=Divisor×7
Now, divide 539 by 7: Divisor=5397=77Divisor = \frac{539}{7} = 77Divisor=7539=77
Answer: 77
Explanation
To determine the number by which 546 should be divided to obtain a quotient of 7 and a remainder of 7, use the fundamental rule of division. This rule expresses a dividend as the product of the divisor and quotient plus the remainder. In algebraic terms, it is written as: Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}Dividend=(Divisor×Quotient)+Remainder
This formula helps relate the four key elements of a division problem: the number being divided (dividend), the number dividing (divisor), the result (quotient), and the leftover part (remainder).
In this case, the dividend is 546. The quotient obtained after division is 7, and the remainder is also 7. Substituting these values into the formula: 546=(Divisor×7)+7546 = (\text{Divisor} \times 7) + 7546=(Divisor×7)+7
To isolate the divisor, begin by subtracting the remainder from the dividend: 546−7=Divisor×7⇒539=Divisor×7546 – 7 = \text{Divisor} \times 7 \Rightarrow 539 = \text{Divisor} \times 7546−7=Divisor×7⇒539=Divisor×7
Now, solve for the divisor by dividing both sides by 7: Divisor=5397=77\text{Divisor} = \frac{539}{7} = 77Divisor=7539=77
Hence, 546 divided by 77 gives a quotient of 7 and a remainder of 7.
This problem emphasizes understanding the structure of division beyond simple computation. It requires recognizing the relationship among the components of division and using algebraic manipulation to find the missing element. Such problems are foundational in building a deeper understanding of arithmetic and algebraic reasoning.
