Ch 05 - Number Play8th Ganita Prakash
Arithmetic Parity Rules: Addition and Subtraction
State the three core parity rules for addition and subtraction. What does this algebraically imply about the relationship between and ?
State the three core parity rules for addition and subtraction. What does this algebraically imply about the relationship between and ?
- Odd Odd: Even
- Even Even: Even
- Odd Even: Odd
Algebraic Implication:
- The expressions and ALWAYS have the exact same parity, regardless of the individual parities of and .
Boundary Conditions / Domain:
- . These parity properties rigorously apply only to integers, not rational fractions or real numbers.
Prism Memory Hook
Like parities make Even, opposite parities make Odd.
Arithmetic Parity Rules: Multiplication
Under what strict mathematical condition does the multiplication of multiple integers yield an odd result?
Under what strict mathematical condition does the multiplication of multiple integers yield an odd result?
- Odd Odd: Odd
- Even Even: Even
- Even Odd: Even
Conclusion:
- A product is even if and only if at least one of its factors is an even number.
Boundary Conditions / Domain:
- The factors must be integers ().
Prism Memory Hook
Even infects everything it multiplies.
Universal Form of Even Algebraic Expressions
Universal Form of Even Algebraic Expressions
State the universal algebraic form that guarantees an expression evaluates to an even number.
Derivation
- Factor out the number from the algebraic terms of the expression.
- E.g., for , factor it as .
- Since integers are closed under addition and multiplication, is an integer. Let .
- The expression takes the rigid form , formally proving it is even.
Component Anatomy:
- : The evaluated expression.
- : Any integer resulting from combining other variables.
Boundary Conditions / Domain:
- Constraint: . If is not an integer (e.g., ), evaluates to (odd), breaking the form.
Prism Memory Hook
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Also in this chapter
29 more cards to study in Prism
- 4Parity Trap: Non-Linear Expressions
- 5Sign Switch Parity Invariance Proof
- 6Parity Invariance of Four Consecutive Numbers
- 7Modulo 4 Classification of Even Numbers
- 8Algebraic Proof: Sum of Multiples of 4
- 9Algebraic Proof: Sum of Two Non-Multiples of 4
- 10Modulo 4 Trap: Sum of Mixed Even Numbers
- 11Representation as Sum of Consecutive Numbers
- 12Trap: Numbers Inexpressible as Consecutive Sums
- 13Theorem: Divisibility of Sums and Differences
- 14Theorem: Divisibility Propagation to Multiples
- 15Transitive Divisibility by Factors
- 16Combined Divisibility (LCM Rule)
- 17Trap: Non-Coprime Combined Divisibility
- 18Divisibility of $n$ Consecutive Integers
- 19Algebraic Foundation of Remainders (Euclidean Division)
- 20Negative Remainder Equivalence
- 21SOP: Simultaneous Remainders (Constant Deficit)
- 22Algebraic Basis: Divisibility by 9
- 23Place Value Face Remainder (Modulo 9)
- 24SOP: Sum of Digits Divisibility Test (9 and 3)
- 25Digital Root: Definition and Modulo 9 Integration
- 26Trap: Digital Roots of Multiples of 3 vs. 9
- 27Digit Shuffle Invariance (Modulo 9 & 3)
- 28Modulo 3: Sum of Three Arbitrary Integers
- 29Algebraic Basis: Divisibility by 11
- 30SOP: Alternating Sum Divisibility Test (11)
- 31Cryptarithms: Core Rules and Constraints
- 32SOP: Solving Multiplication Cryptarithms (Units Analysis)
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8th Ganita Prakash
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