Divisibility Criteria | Traditional Summary
Contextualization
Divisibility is a fundamental concept in mathematics that allows us to determine whether a number can be divided by another without leaving a remainder. This knowledge is essential both for solving mathematical problems and for practical applications in everyday life. For example, when splitting a bill among friends or organizing objects into equal groups, understanding the criteria for divisibility helps us perform these tasks efficiently and accurately.
The criteria for divisibility are simple rules that allow us to quickly verify whether a number is divisible by another, without the need to perform the full division. In this lesson, we will focus on the criteria for divisibility by 2, 3, 4, 5, 6, 9, and 10. These criteria not only facilitate the resolution of mathematical problems but are also widely used in various areas, such as computer programming, where checking for divisibility by 2 is crucial for the functioning of the binary systems used by computers.
Divisibility Criterion for 2
A number is divisible by 2 if it is an even number. In other words, a number is divisible by 2 if its last digit is one of the following: 0, 2, 4, 6, or 8. This rule is quite simple and straightforward, making it easy to apply in any situation.
For example, consider the numbers 14, 22, and 30. All of these numbers have a final digit that is even (4, 2, and 0, respectively), so we know they are divisible by 2. This means that if we divide any of these numbers by 2, the result will be an integer, without a remainder.
This criterion is widely used in various fields, including computer programming, where the need to check the parity of numbers is common. In the binary system, which is the foundation upon which computers operate, divisibility by 2 is a fundamental operation.
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A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
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Numbers divisible by 2 are called even numbers.
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This criterion is widely used in programming and computing.
Divisibility Criterion for 3
A number is divisible by 3 if the sum of its digits is divisible by 3. This criterion requires a bit more calculation than the criterion for divisibility by 2, but it is still quite simple to apply.
For example, consider the number 123. To check if it is divisible by 3, we add its digits: 1 + 2 + 3 = 6. Since 6 is divisible by 3, we can conclude that 123 is also divisible by 3.
The applicability of this criterion can be seen in various contexts, such as in the simplification of fractions and the resolution of mathematical problems involving multiples and divisors. Additionally, understanding divisibility by 3 can be useful in various scientific and engineering fields, where the analysis of numbers and patterns is frequent.
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A number is divisible by 3 if the sum of its digits is divisible by 3.
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This criterion is useful in the simplification of fractions and the resolution of mathematical problems.
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Applicable in various scientific and engineering fields.
Divisibility Criterion for 5
A number is divisible by 5 if its last digit is 0 or 5. This is one of the easiest criteria to remember and apply, as it only depends on observing the last digit of the number.
For example, the numbers 25, 50, and 75 are all divisible by 5, as their last digits are 5, 0, and 5, respectively. This means that when dividing any of these numbers by 5, we will obtain an integer without a remainder.
This criterion is particularly useful in practical everyday situations, such as when counting money (where coins and bills often come in multiples of 5) or measuring time (since many time intervals are measured in multiples of 5 minutes).
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A number is divisible by 5 if its last digit is 0 or 5.
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This criterion is easy to remember and apply.
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Useful in practical situations such as counting money and measuring time.
Divisibility Criterion for 10
A number is divisible by 10 if its last digit is 0. This criterion is extremely simple and straightforward, as it only depends on observing the last digit of the number.
For example, consider the numbers 40, 70, and 100. All of these numbers end in 0, which means they are divisible by 10. This also means that when dividing these numbers by 10, the result will be an integer without a remainder.
This rule is very practical in various everyday situations, such as when working with measurements (where many measurement systems are based on multiples of 10) or when counting and grouping items in packs of 10. Furthermore, divisibility by 10 is a fundamental concept in arithmetic and in understanding decimal numbering systems.
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A number is divisible by 10 if its last digit is 0.
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This criterion is easy to observe and apply.
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Very practical in everyday situations and fundamental in arithmetic.
To Remember
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Divisibility: The ability of a number to be divided by another without leaving a remainder.
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Even Number: A number that is divisible by 2.
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Sum of Digits: The addition of all the digits of a number.
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Integer: A number without fractional or decimal parts.
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Binary System: A base 2 numbering system used by computers.
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Multiple: A number that can be divided by another number without leaving a remainder.
Conclusion
In this lesson, we explored the main criteria for divisibility for the numbers 2, 3, 4, 5, 6, 9, and 10. We learned that these criteria allow us to quickly verify whether a number is divisible by another without the need to perform the full division. This is extremely useful for both solving mathematical problems and for practical applications in everyday life.
Understanding the criteria for divisibility helps us simplify calculations and solve problems more efficiently. Furthermore, these criteria are fundamental in various areas of mathematics and their applications extend to other fields, such as computing and engineering. For example, divisibility by 2 is crucial for the functioning of the binary systems used by computers.
The knowledge gained in this lesson not only facilitates the understanding and application of mathematical concepts but also prepares students to face challenges in different contexts. We encourage everyone to explore more about the subject and to practice the criteria for divisibility, as constant practice is essential for consolidating learning.
Study Tips
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Practice regularly with exercise lists that include different criteria for divisibility. This will help to solidify the concepts and develop problem-solving skills.
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Review the practical examples discussed in class and try to create your own examples. This will help internalize the criteria and apply them in various situations.
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Use additional resources, such as educational videos and online math games, to make studying more dynamic and interesting. These resources can offer different perspectives and teaching methods that complement learning.
