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Chapter 1: Problem 49

Multiply and reduce to lowest terms. $$ 25 \cdot 318 \cdot 45 $$

Short Answer

Expert verified
The product is 357750 and doesn't require further simplification.

Step by step solution

01

Multiply the First Two Numbers

First, we will multiply the numbers 25 and 318. The multiplication of these numbers is:\[25 \times 318 = 7950\]
02

Multiply the Result with the Third Number

Now, we will multiply the result from Step 1, which is 7950, with the third number, 45. The multiplication is:\[7950 \times 45 = 357750\]
03

Reduce to the Lowest Terms

There isn't an inherent simplification needed further for a product. However, if this was meant to be simplified as a fraction (e.g., factor 357750), check common factors. Let's reduce 357750 using common factors. The greatest common divisor for simplification isn't necessary here since the product itself is the end calculation, but finding it may not change the integer result further. As an arithmetic value and depending on context, simplest reporting may just be the product.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic
Arithmetic deals with the fundamental operations of numbers: addition, subtraction, multiplication, and division. It's the foundation of all mathematical calculations. Every arithmetic operation involves two or more numbers combined in a specific manner. For multiplication, which this exercise centers around, it involves taking a number and adding it to itself a set number of times. In our case, multiplying 25 by 318 means adding 25, 318 times. Similarly multiplying this result with another number, 45 here, requires repeating this new total 45 more times. This rigorous process reinforces arithmetic's core role in math.
Greatest Common Divisor
The greatest common divisor (GCD) is crucial in reducing numbers to their lowest terms, especially fractions. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder.
In our exercise, although the GCD isn't initially necessary because we're dealing with a product (357750), understanding this concept is crucial whenever simplification is needed. If we had to express this product in fractional terms or needed a more reduced form in other contexts, finding the GCD of 357750 with another number could aid in simplification.
Product
A product is the result of multiplying numbers together. This operation is a cornerstone of multiplication, producing a single number from two or more factors. In the exercise, we multiplied three numbers:
  • First, 25 and 318 to get 7950.
  • Then, 7950 and 45 to get 357750.
This final value, 357750, represents the product of the original numbers listed in the exercise. Each step consolidates smaller multiplicative pairs into a larger product, exemplifying how products arise in arithmetic tasks.
Integer Result
An integer result refers to a final output that is a whole number, not a fraction or decimal. In our scenario, after performing all multiplications, the result is 357750, which is a clean, whole number or integer. This is significant because many arithmetic outcomes in real-world applications require integer results.
Additionally, when reducing numbers to their simplest form, maintaining an integer outcome often remains a priority. Such an approach is especially useful when dealing with countable quantities, where fractions would not make practical sense.

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