Mathematics
Grade 11
15 min
Divisibility rules for 4 and 8
Divisibility rules for 4 and 8
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1
Introduction & Learning Objectives
Learning Objectives
Algebraically represent any integer in base-10.
Prove the divisibility rules for 4 and 8 using polynomial expansion.
Apply the divisibility rules for 4 and 8 to expressions containing variables.
Determine the possible values of an unknown digit in a number to satisfy divisibility by 4 or 8.
Analyze the divisibility of an entire algebraic expression, not just a single term.
Construct arguments about number properties using modular arithmetic concepts (e.g., N ≡ 0 (mod 4)).
How can you tell if a massive number like 583,497,13_ is divisible by 8 without a calculator, even with a missing digit? 🤔 Let's find out!
This tutorial moves beyond simple arithmetic checks. We will explore the algebraic foundations of the divisibility rules for 4 and 8, applying...
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Key Concepts & Vocabulary
TermDefinitionExample
DivisibilityAn integer 'a' is divisible by a non-zero integer 'b' if there exists an integer 'k' such that a = bk. This is equivalent to saying that the remainder of a ÷ b is 0.24 is divisible by 8 because 24 = 8 * 3, where k=3.
Base-10 RepresentationThe standard system for writing numbers, where a number is expressed as a sum of terms, each being a digit multiplied by a power of 10.The number 5,283 can be written as the variable expression 5(10^3) + 2(10^2) + 8(10^1) + 3(10^0).
Variable ExpressionA mathematical phrase that contains numbers, variables (like x or n), and operators (+, -, *, /). In this context, we use them to represent numbers with unknown digits or general forms of integers.The expression 100a + 10b + c represents any th...
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Core Formulas
Algebraic Rule for Divisibility by 4
An integer N is divisible by 4 if and only if the number formed by its last two digits is divisible by 4. For N = 100k + (10t + u), N is divisible by 4 if and only if (10t + u) is divisible by 4.
Since 100 is divisible by 4 (100 = 4 * 25), any multiple of 100 is also divisible by 4. Therefore, we only need to check the divisibility of the part of the number that is less than 100, which is the number formed by the tens (t) and units (u) digits.
Algebraic Rule for Divisibility by 8
An integer N is divisible by 8 if and only if the number formed by its last three digits is divisible by 8. For N = 1000k + (100h + 10t + u), N is divisible by 8 if and only if (100h + 10t + u) is divisible by 8.
Since 1000 is divisible by 8 (1000 = 8 * 125), any...
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Challenging
An integer N is represented as 1000a + 100b + 10c + d. Given that N is divisible by 8, which of the following expressions MUST evaluate to an integer?
A.(b+c+d)/8
B.(10c+d)/8
C.(4b+2c+d)/8
D.(100b+10c+d)/4
Easy
An integer N is represented in base-10 as N = 100k + (10t + u), where k, t, and u are integers representing parts of the number. According to the algebraic rule for divisibility by 4, which condition must be true for N to be divisible by 4?
A.k must be divisible by 4.
B.The expression (10t + u) must be divisible by 4.
C.The sum k + t + u must be divisible by 4.
D.The digit u must be divisible by 4.
Easy
An integer N is represented as N = 1000k + (100h + 10t + u). The algebraic proof for the divisibility rule for 8 relies on which property of the base-10 system?
A.10 is divisible by 2.
B.100 is divisible by 4.
C.1000 is divisible by 8.
D.The sum of the digits determines divisibility.
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