Mathematics
Grade 11
15 min
Divisibility rules for 2, 5, and 10
Divisibility rules for 2, 5, and 10
What you'll learn
- Identify if a number less than 100 is divisible by 2 by checking if the last digit is 0, 2, 4, 6, or 8 with 90% accuracy.
- Explain why a number ending in 0 or 5 is divisible by 5 using examples.
- Apply the divisibility rule of 10 to determine if a number less than 100 is divisible by 10 by stating whether the last digit is a 0.
- Solve 5 out of 6 problems correctly, determining if a given number (less than 100) is divisible by 2, 5, or 10.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Algebraically prove why the divisibility rules for 2, 5, and 10 work using polynomial representations of integers.
Analyze a polynomial expression P(n) and determine the conditions on the integer variable 'n' that make P(n) divisible by 2, 5, or 10.
Decompose variable expressions into components that are demonstrably divisible by 2, 5, or 10 and a remaining component to be analyzed.
Construct variable expressions that are always divisible by 2, 5, or 10 for any integer input.
Apply the concept of divisibility to solve problems involving sequences and series where terms must meet specific criteria.
Evaluate the divisibility of complex algebraic expressions by identifying the controlling term(s).
How can we predict if a complex algebraic formula,...
2
Key Concepts & Vocabulary
TermDefinitionExample
DivisibilityAn integer 'a' is divisible by a non-zero integer 'b' if there exists an integer 'c' such that a = bc. In the context of expressions, a variable expression P(n) is divisible by 'b' if P(n) = b * Q(n) for some expression Q(n) that yields an integer for any integer 'n'.The expression 10n + 5 is divisible by 5 because it can be factored as 5(2n + 1). Since 'n' is an integer, '2n+1' is also an integer.
Polynomial Representation of an IntegerAny integer can be expressed as a sum of powers of 10. This is the basis of our decimal system and is crucial for proving divisibility rules algebraically.The number 483 can be written as 4 * 10^2 + 8 * 10^1 + 3 * 10^0. We can generalize this as N = a_k...
3
Core Formulas
Algebraic Rule for Divisibility by 2
An expression P(n) is divisible by 2 if P(n) \equiv 0 \pmod{2}. For any integer N = 10k + d_0, its divisibility by 2 is determined entirely by its last digit, d_0, because 10k is always divisible by 2.
To check if a variable expression is divisible by 2, decompose it into terms that are always even (e.g., multiples of 2, 10, etc.) and analyze the remaining terms. The expression is divisible by 2 if and only if the sum of the remaining terms is divisible by 2.
Algebraic Rule for Divisibility by 5
An expression P(n) is divisible by 5 if P(n) \equiv 0 \pmod{5}. For any integer N = 10k + d_0, its divisibility by 5 is determined entirely by its last digit, d_0, because 10k is always divisible by 5.
To check if a variable expression is divisibl...
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Easy
According to the algebraic definition of divisibility for variable expressions, the expression P(n) is divisible by 5 if and only if which of the following conditions is met for some expression Q(n) that yields an integer for any integer n?
A.P(n) + 5 = Q(n)
B.P(n) / Q(n) = 5n
C.P(n) = 5 * Q(n)
D.5 * P(n) = Q(n)
Easy
The algebraic proof for the divisibility rule of 2 for any integer N = 10k + d₀ relies on a key property of the term 10k. What is this property?
A.10k is always an odd number.
B.10k is always divisible by 2.
C.10k is only divisible by 2 if k is even.
D.10k is always divisible by 4.
Easy
When analyzing the expression P(n) = 10n² + 3n + 5 to determine its divisibility by 5, which term's divisibility is not guaranteed for all integer values of n?
A.10n²
B.5
C.10n² + 5
D.3n
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Frequently asked questions
What grade level is "Divisibility rules for 2, 5, and 10"?
Divisibility rules for 2, 5, and 10 is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Divisibility rules for 2, 5, and 10?
You'll be able to: Identify if a number less than 100 is divisible by 2 by checking if the last digit is 0, 2, 4, 6, or 8 with 90% accuracy; Explain why a number ending in 0 or 5 is divisible by 5 using examples; Apply the divisibility rule of 10….
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How many practice questions are included with Divisibility rules for 2, 5, and 10?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.