Problem 25 - Fundamentals of Mathematics

Question

Does 7 divide into \(5^{2} \cdot 6^{3} \cdot 7^{4} \cdot 8 ?\) Explain.

Step-by-Step Solution

Answer

Yes, the expression is divisible by 7 as it includes the factor \(7^{4}\).

Step 1: Understand the Problem

We need to determine if the number \(5^{2} \cdot 6^{3} \cdot 7^{4} \cdot 8\) is divisible by 7. This means checking if the number, when divided by 7, leaves a remainder of 0.

Step 2: Identify the Contribution of 7

In the expression \(5^{2} \cdot 6^{3} \cdot 7^{4} \cdot 8\), notice that it includes \(7^{4}\). This means that the number already has 7 as a factor.

Step 3: Simplify the Expression Considering Divisibility by 7

Since \(7^{4}\) is clearly divisible by 7, we can conclude that the entire product \(5^{2} \cdot 6^{3} \cdot 7^{4} \cdot 8\) is divisible by 7. The factor \(7^{4}\) ensures that the product is divisible by 7, as all other numbers do not affect divisibility by 7.

Step 4: Conclusion

Since \(7^{4}\) is present in the multiplication, and any number which includes a factor of a multiple of 7 will be divisible by 7, the entire expression is divisible by 7. Therefore, 7 divides into \(5^{2} \cdot 6^{3} \cdot 7^{4} \cdot 8\) without a remainder.

Key Concepts

Understanding Factors

Factors are the building blocks of numbers. They are integers that can be multiplied together to produce another number. For example, factors of 12 are 1, 2, 3, 4, 6, and 12 because these numbers divide 12 without leaving a remainder.
In the context of the exercise, identifying a factor of 7 within the expression means the whole expression is divisible by 7, if the factor 7 is raised to a non-zero power. This is the case with the expression \(7^4\), indicating that 7 is indeed a factor.

Role of Multiplication in Divisibility

Multiplication plays a significant role in understanding divisibility. In this exercise, the expression \(5^2 \cdot 6^3 \cdot 7^4 \cdot 8\) results from multiplying several numbers.
Each factor contributes to a part of the multiplication product. Because of multiplication, if just one segment of the product is divisible by a number, it ensures the divisibility of the entire expression.

• Here, since \(7^4\) forms a part of the product, and \(7^4\) itself contains the factor 7, the entire expression is confirmed to be divisible by 7.

Applying Mathematical Reasoning

Mathematical reasoning involves logical thinking to solve problems. It helps us deduce truths based on given patterns and properties.
In examining the given expression \(5^2 \cdot 6^3 \cdot 7^4 \cdot 8\), using mathematical reasoning means recognizing that because it includes \(7^4\), 7 is inherently a factor.
This immediately leads to the realization that the full multipication is divisible by 7.
Mathematical reasoning simplifies the verification process significantly and avoids unnecessary complex computations.

Exploring Number Theory

Number theory focuses on the properties and relationships of numbers, especially integers. This field of study furnishes tools to explore primers like divisibility rules.
In this exercise, the presence of 7 in the power \(7^4\) facilitates an application of number theory. Since number theory tells us that any expression containing a base raised to an exponent is divisible by that base, we can conclude the expression is divisible by 7.

• Furthermore, number theory helps us quickly identify and utilize divisibility criteria, which in this case confirms our final solution.