Question

Is $n$ divisible by 8 with no remainder? (1) $\mathrm{n}+8$ is divisible by 8 with integer result (2) $\mathrm{n}+16$ is divisible by 8 with integer result A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

D. Either clue 1 or clue 2 alone is sufficient.

Step-by-step solution

- Understand the problem

We need to determine whether the integer n is divisible by 8 with no remainder. We are given two clues: (1) n+8 is divisible by 8, and (2) n+16 is divisible by 8. We need to evaluate these clues individually and together.

- Evaluate clue 1

If n+8 is divisible by 8, it means that n+8 = 8k for some integer k. Simplifying this, we get n = 8k - 8, which means n is 8 times (k-1), confirming that n is divisible by 8.

- Evaluate clue 2

If n+16 is divisible by 8, it means that n+16 = 8m for some integer m. Simplifying this, we get n = 8m - 16, which means n is 8 times (m-2), confirming that n is divisible by 8.

- Combine the clues

Since both clues individually show that n is divisible by 8, combining them doesn't provide any additional information. Therefore, either clue alone is sufficient to determine that n is divisible by 8.

- Answer the question

The correct answer is D. Either clue 1 or clue 2 alone is sufficient to determine that n is divisible by 8.

Key concepts

Divisibility Rules

Divisibility rules help us determine if one number can be divided by another without leaving a remainder. For example, a number is divisible by 8 if the last three digits of the number form a value that is divisible by 8. These rules simplify the division process and save time on calculations. In our exercise, if we see that both terms like $n+8$ and $n+16$ result in multiples of 8, then $n$ must be inherently divisible by 8. This verification can help students quickly solve problems without performing long division.

Logical Reasoning

Logical reasoning is crucial for solving math problems effectively. It involves breaking down problems into manageable parts and using given clues to derive conclusions. For instance, in our problem, noticing that $n+8$ and $n+16$ are divisible by 8 suggests that both clues independently lead to the result that $n$'s divisibility by 8 must be true. Each clue by itself is enough to conclude the same result. Understanding how to piece together information logically ensures accuracy and efficiency in problem-solving.

Integer Properties

Integer properties help us understand and manipulate whole numbers. In our problem, integers $k$ and $m$ are used to express $n$'s divisibility. The clues state that $n+8=8k$ and $n+16=8m$, implying $n=8(k-1)$ and $n=8(m-2)$ respectively. Since $k-1$ and $m-2$ are integers, it follows that $n$ is a multiple of 8. Properties such as these make it easier to deduce characteristics of $n,$ as they provide clear, integer-based frameworks for understanding relationships between numbers.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a statement. In this exercise, we use proof by deduction. Starting from the clues $n+8=8k$ and $n+16=8m$, we rearrange these to show $n=8(k-1)$ and $n=8(m-2)$, respectively. Since $k-1$ and $m-2$ are integers, $n$ must be divisible by 8. These steps form a mathematical proof, providing a reliable solution to the problem. Proofs build a solid understanding of how and why mathematical principles hold true.