Question

If \(x\) is the remainder when a multiple of 4 is divided by \(6,\) and \(y\) is the remainder when a multiple of 2 is divided by \(3,\) what is the greatest possible value of \(x+y ?\) A. $$2$$ B. $$3$$ C. $$5$$ D. $$6$$ E. $$9$$

Short answer

The greatest possible value of \(x+y\) is 6. Hence, the answer is (D) 6.

Step-by-step solution

Understanding Remainders

Whenever a number is divided by another, the remainder can never be equal or larger than the divisor. So what is the multiple of 4 divided by 6 and the multiple of 2 divided by 3, the remainders \(x\) and \(y\) respectively, will always be less than 6 and 3.

Calculate Remainder \(x\)

For any multiple of 4, when divided by 6, the greatest possible remainder is 4. For example, when you divide 4 (multiple of 4) by 6, the remainder \(x\) is 4. Because 6 goes into 4 zero times with a remainder of 4.

Calculate Remainder \(y\)

For any multiple of 2, when divided by 3, the greatest possible remainder is 2. For example, when you divide 2 (a multiple of 2) by 3, the remainder \(y\) is 2. Because 3 goes into 2 zero times with a remainder of 2.

Add Remainders

Finally, the sum of \(x+y\) will be the greatest possible value. Adding the remainder values: \(x (4) + y (2) = 6\)

Key concepts

Remainders

When a number is divided by another number, it may not always divide evenly. The leftover or what remains after the division is called the remainder. For example, when we divide 14 by 5, the quotient is 2 and the remainder is 4, since 5 fits into 14 twice with a leftover of 4.
Remainders apply when dealing with integers, and the remainder is always smaller than the divisor. This means if you're dividing by 6, the remainder could be 0, 1, 2, 3, 4, or 5, but never 6 or larger. Understanding remainders is crucial in solving problems involving divisibility and number theory.

Divisibility Rules

Divisibility rules are shortcuts that help us determine if one number divides evenly into another, without performing the actual division. For instance, a number is divisible by 4 if the last two digits form a number divisible by 4. Divisibility rule for 2 is simple; a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. This helps when solving remainder problems.
In this problem, understanding the divisibility rules for 4 and 2 guides us in finding multiples that, when divided by other numbers, give the remainders. By applying these rules, determining the greatest possible value of remainders in such scenarios becomes easier.

Mathematical Reasoning

Mathematical reasoning is the logical thought process required to understand and solve problems. It involves breaking down complex problems into simpler parts and using known mathematical concepts to reach a conclusion. In exercises such as the one given, mathematical reasoning allows us to understand how remainders work and how to manipulate them correctly.
By reasoning through the problem steps: identifying possible remainders for each divisor and then summing those remainders logically, we reach the final solution. This logical approach ensures that solutions are robust and accurate.

Number Theory

Number theory is a field of pure mathematics concerned with the properties and relationships of numbers, particularly integers. It covers concepts like divisibility, the behavior of integers, and modular arithmetic, all of which are crucial for solving the given problem.
Within this context, number theory helps us understand how numbers behave when divided and what patterns emerge. By recognizing that multiples offer predictable remainder patterns, like how multiples of 4 behave when divided by 6, we can solve for the greatest possible sum of remainders efficiently. Number theory's deep dive into such properties is what makes finding solutions both interesting and insightful.