Problem 61 - Algebra 2

Question

State whether each statement is true or false when \(n=1\). Explain. \(7^{n}-3^{n}\) is divisible by 4

Step-by-Step Solution

Answer

True, because \( 7^1 - 3^1 = 4 \) and 4 is divisible by 4.

Step 1: Substitute the value of n

First, substitute the given value of \( n = 1 \) into the expression \( 7^n - 3^n \) to evaluate it. This results in \( 7^1 - 3^1 \).

Step 2: Simplify the expression

Calculate \( 7^1 \) and \( 3^1 \). We have:\[7^1 = 7\]\[3^1 = 3\]Now, substitute these values into the expression to get:\[7 - 3 = 4\]

Step 3: Check divisibility by 4

Since the result from Step 2 is 4, we need to check if 4 is divisible by 4. Dividing 4 by 4 gives a quotient of 1 with a remainder of 0, meaning 4 is indeed divisible by 4.

Step 4: State the final verdict

Because \( 7^1 - 3^1 = 4 \) and 4 is divisible by 4, we can conclude that the statement is true when \( n = 1 \).

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication compactly. When you see an expression like \(7^n\), it means that the number 7 is multiplied by itself \(n\) times. For example, when \(n = 1\), \(7^n\) simplifies to just 7, as 7 multiplied by itself once is 7. Similarly, \(3^n\) becomes just 3 when \(n = 1\). It's important to understand how exponents work because they allow us to handle large calculations more easily by using the base and exponent notation.
Exponents have several key properties that make them useful in algebraic expressions, such as the product of powers (which states \(a^m \times a^n = a^{m+n}\)), and the power of a power (which states \((a^m)^n = a^{m\times n}\)). These properties are handy when simplifying expressions where exponents are involved.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In our exercise, the expression \(7^n - 3^n\) consists of two exponential terms, \(7^n\) and \(3^n\), and an operation of subtraction. These expressions can change in value based on the substitutions made for any variables they contain.
Substituting values for variables is a common technique in algebra to evaluate expressions. In this particular example, setting \(n = 1\) transforms the algebraic expression into a simpler arithmetic expression that we can easily solve: \(7^1 - 3^1\). Understanding how to manipulate and evaluate algebraic expressions is fundamental to solving algebra problems and making sense of numerical relationships.

Integer Division

Integer division is the process of dividing one integer by another, yielding a quotient and possibly a remainder. In the context of our exercise, we looked at the expression \(7^1 - 3^1 = 4\) to determine whether it was divisible by 4. To check divisibility, divide 4 by 4, which equals 1 with no remainder.
This means that 4 is completely divisible by 4. Divisibility without any remainder shows a clear integer division, confirming that our original problem's statement is true for \(n = 1\). Knowing how integer division works, and how to check divisibility, is beneficial for understanding the nature of numbers and their properties in algebra and number theory.