Question

Is \(\mathrm{n}\) divisible by 4 with no remainder? (1) \(\mathrm{n}^2\) is divisible by 4 with integer result (2) \(\mathrm{n}^2+4 \mathrm{n}\) is divisible by 16 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

B. 2 alone, not 1 alone

Step-by-step solution

- Analyze Statement 1

Statement 1 says that \(^2 \) is divisible by 4 with an integer result. If \(^2 \) is divisible by 4, then \(n \) also has to be divisible by 2, since \(2^2 = 4\). To determine if \(n \) is divisible by 4, \(n = 2k \) can be checked to see if \(n \) has another factor of 2.

- Analyze Statement 2

Statement 2 says that \(n^2 + 4n \) is divisible by 16. Factor the expression: \(n^2 + 4n = n(n + 4)\). Suppose \(n = 4k + r \), for \(r \) being the remainder, only values \(0, 1, 2, 3\) for \(r \) need checking. Substitute these into \((4k + r)(4k + r + 4) \); \(r = 0 \) results in it always divisible by 16.

- Combine Analysis

From Steps 1 and 2, statement 1 suggests that \(n \) being a multiple of 2 doesn't guarantee it being a multiple of 4. However, statement 2 shows confirming \(n^2 + 4n \) divisible by 16 guarantees \(n \) is indeed multiple of 4. Therefore, statement 2 is more conclusive.

Key concepts

Divisibility Rules

Divisibility rules are simple tests to determine if one number is divisible by another without performing division. For instance, a number is divisible by 4 if the number formed by its last two digits is divisible by 4. Knowing these rules helps quickly identify factors of numbers without lengthy calculations. Applying these rules to the exercise, we see that if \(n^2\) is divisible by 4, then \(n\) must be divisible by 2 because squaring a number preserves its divisibility properties. Similarly, the expression \(n^2 + 4n\) being divisible by 16 assures that certain conditions about \(n\) need to be met, which stems from understanding and applying divisibility rules.

Mathematical Proof

Mathematical proof is a logical argument demonstrating that a specific statement or theorem is true. It uses previously established facts, definitions, and axioms. In the given exercise, we use proof by direct calculation. First, we assume statements about \(n^2\) and \(n^2 + 4n\). We then check these against known properties of numbers. For example, we see if substituting values into the factorized form of \(n^2 + 4n = n(n + 4)\) holds true for divisibility by 16. This methodically shows if divisibility by 4 is a necessary consequence.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. They are foundational in forming equations and functions. In our exercise, we work with expressions like \(n^2\) and \(n^2 + 4n\). Understanding how to manipulate these expressions—factoring, expanding, and simplifying—is crucial. For example, \(n^2 + 4n\) is factored to \(n(n + 4)\). This reveals that for it to be divisible by 16, both factors together must align in a specific way, leading to insights about \(n\).

Integer Properties

Integer properties cover various aspects like even/odd distinctions, positive/negative values, and divisibility. In analyzing the problem, we notice that if \(n^2\) is divisible by 4, \(n\) must be even. Further, if \(n^2 + 4n\) is divisible by 16, certain integer properties confirm \(n\) must be a multiple of 4. By substituting \(n = 2k\) (where k is an integer), and testing for further divisibility, these properties guide our proof that \(n\), being divisible by 4, fulfills the conditions set by the problem's statements.