Question

When a number \(^{+} N^{\prime}\) is divided by a proper divisor ' \(D^{\prime}\) thenj leaves a remainder of 14 and if the thrice of that numb i.e., \(3 N\) is divided by the same divisor \(D\), the remaindes comes out to be 8. Again if the 4 times of the same number i.e., ' \(4 N^{\prime}\) is divided by \(D\) the remainder will be : (a) 35 (b) 22 (c) 5 (d) can't be determined

Short answer

Based on the given information and step-by-step solution, we cannot determine a unique value for the remainder when 4 times the number is divided by the given divisor. Therefore, the answer is: (d) can't be determined.

Step-by-step solution

Write down the given equations representing the remainders

We are given that when \(N'\) is divided by \(D'\), the remainder is 14. This can be represented as: $$N' = D' \cdot Q_1 + 14$$ where \(Q_1\) is the quotient. Similarly, when \(3N\) is divided by \(D\), the remainder is 8. This can be represented as: $$3N = D' \cdot Q_2 + 8$$ where \(Q_2\) is the quotient. Finally, we need to find the value of the remainder when \(4N'\) is divided by \(D\). This can be represented as: $$4N' = D' \cdot Q_3 + R$$ where \(Q_3\) is the unknown quotient and \(R\) is the unknown remainder.

Combine the two given equations

We want to find the relationship between \(4N'\) and \(D'\). For that, we can try to express \(4N'\) in terms of \(D'\) and the given equations. From the first equation, we have: $$N' = D' \cdot Q_1 + 14$$ Now, find \(4N'\): $$4N' = 4(D' \cdot Q_1 + 14)$$

Express the remainder in terms of D'

Now, let's substitute the expression of \(3N\) from the second equation into the above equation: $$3N = D' \cdot Q_2 + 8$$ As we know, \(3N = N' + 2N'\), we can rewrite the second given equation as: $$N' + 2N' = D' \cdot Q_2 + 8$$ So, $$2N' = (D' \cdot Q_2 + 8) - N'$$ Now, multiply the expression by 2: $$4N' = 2(D' \cdot Q_2 + 8) - 2N'$$

Solve for the unknown remainder R

We now have an expression for \(4N'\) in terms of \(D'\) that we can substitute back into the equation for \(4N'\): $$4N' = D' \cdot Q_3 + R$$ From previous steps, the equation for \(4N'\) is: $$4N' = 4(D' \cdot Q_1 + 14) = 2(D' \cdot Q_2 + 8) - 2N'$$ Now we can plug the expression for \(4N'\) into the remainder equation: $$2(D' \cdot Q_2 + 8) - 2N' = D' \cdot Q_3 + R$$ From this point, we have a single equation, but we still have three unknowns: \(Q_1, Q_2,\) and \(Q_3\). Since there are no constraints on these unknowns, we cannot determine a unique solution for the remainder.

Conclude the answer

Since we cannot determine a unique solution for the remainder, the correct answer is: (d) can't be determined

Key concepts

Divisibility Rules

Understanding the rules of divisibility can greatly simplify numerous problems in number theory, such as finding remainders. The essence of divisibility is determining whether a number is evenly divisible by another without performing the actual division. For example, the rule for divisibility by 2 states that if the last digit of a number is even (i.e., 0, 2, 4, 6, or 8), then the number is divisible by 2.

Similarly, a number is divisible by 3 if the sum of its digits is divisible by 3. These rules are shortcuts that help us to find remainders and quotients more quickly. For instance, knowing that a number ending in 5 or 0 is divisible by 5 allows us to determine that when a number ending in 14 is divided by 5, the remainder must be 4. The utility of divisibility rules becomes even more evident when working with larger numbers or when trying to find factors or multiples.

Remainder Theorem

The Remainder Theorem is a powerful tool in number theory which relates the remainder obtained when a polynomial is divided by a linear divisor. It states that if a polynomial function f(x) is divided by a linear divisor of the form x - k, the remainder is simply f(k). This theorem can be applied to understand remainders in a numerical context as well.

In the context of the provided exercise, the Remainder Theorem could be leveraged to express the relationship between the dividend, the divisor, and the remainder. When given that division of a number by its proper divisor leaves a specific remainder, it is akin to knowing the value of f(k) in the polynomial division scenario. This relation could potentially lead to finding a pattern or relationship which could conclude to the final remainder when the number is multiplied and then divided by the same divisor. The challenge, however, arises when there are insufficient constraints to uniquely determine the remainder, as in our exercise.

Mathematical Reasoning

Mathematical reasoning is the process of using logic and critical thinking skills to solve mathematical problems. It encompasses a broad range of skills from understanding and formulating conjectures to creating and following a coherent chain of arguments that lead to a conclusion. In the textbook problem presented, mathematical reasoning is used to deduce the relationship between different algebraic expressions and remainders.

This process requires an understanding of the properties and operations of integers, variable manipulation, and the ability to identify patterns or formulate general rules. In practice, while working through a problem, reasoning allows us to infer the next logical step or to recognize when a problem might have insufficient data for a conclusive answer—as in the case of our exercise where, despite the steps taken to solve for the unknown remainder, the conclusion is that the remainder can't be determined given the current information.