2.2 Remainders, Even/Odd Parity, and Positive/Negative Sign Analysis

2.2 Remainders, Even/Odd Parity, and Positive/Negative Sign Analysis

Quick Summary: Remainder arithmetic, even/odd parity transformations, and positive/negative sign tracking form the structural bedrock of high-scoring GMAT number properties questions. Recognizing that remainder relationships follow linear algebraic models ($N = dq + r$) and that parity properties instantly constrain sums and products prevents tedious trial-and-error under time pressure.

Quantitative Reasoning on the GMAT Focus Edition repeatedly tests your ability to make logical deductions about expressions without calculating their numerical values. Remainder arithmetic, parity properties, and sign rules provide algebraic shortcuts that immediately eliminate impossible answer choices.

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The Division Algorithm and Remainder Mechanics

The Division Algorithm is a foundational theorem of integer arithmetic. It states that for any integer $N$ (the dividend) and positive integer divisor $d$, there exist unique integers $q$ (the quotient) and $r$ (the remainder) such that:

$N = d \cdot q + r \quad \text{where} \quad 0 \le r < d$

Essential Remainder Principles on the GMAT

1. The Remainder is Strictly Bounded: The remainder $r$ must be an integer satisfying $0 \le r < d$. A remainder can never be negative, nor can it equal or exceed the divisor $d$. For example, when dividing by 7, the only possible remainders are ${0, 1, 2, 3, 4, 5, 6}$.
2. When the Dividend is Less than the Divisor: If $0 < N < d$, then the quotient is 0 and the remainder is $N$ itself. For example, when 5 is divided by 8: $5 = 8(0) + 5 \implies \text{remainder} = 5$
3. Algebraic Form of Sets with a Fixed Remainder: An integer that leaves a remainder of 3 when divided by 5 can be represented as: $N = 5k + 3 \quad (k \in \mathbb{Z})$ For non-negative values of $k$, this yields the sequence ${3, 8, 13, 18, 23, 28, \dots}$.

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Modular Remainder Arithmetic and Cyclicity

When evaluating remainders of large calculations, you can distribute the remainder operation across addition, subtraction, and multiplication:

• Sum Rule: $\text{Rem}(A + B, d) = \text{Rem}(\text{Rem}(A, d) + \text{Rem}(B, d), d)$
• Product Rule: $\text{Rem}(A \times B, d) = \text{Rem}(\text{Rem}(A, d) \times \text{Rem}(B, d), d)$
• Power Rule: $\text{Rem}(A^n, d) = \text{Rem}((\text{Rem}(A, d))^n, d)$

Units Digit Cyclicity (Remainders Modulo 10)

The units digit of any integer is mathematically identical to its remainder when divided by 10. The units digits of positive integer powers cycle in predictable patterns of length 1, 2, or 4:

Shortcut to Find the Units Digit of $b^n$:

1. Divide the exponent $n$ by the cycle length (4 for bases ending in 2, 3, 7, 8).
2. If the remainder is 1, take the 1st power in the cycle.
3. If the remainder is 2, take the 2nd power.
4. If the remainder is 3, take the 3rd power.
5. If the remainder is 0 (exponent is a multiple of 4), take the 4th power in the cycle.

Combined Remainder Systems

A classic GMAT question type specifies remainders across two different divisors: for example, $N$ leaves remainder 2 when divided by 5, and remainder 3 when divided by 7.

Step-by-Step Resolution:

1. Express $N$ algebraically under the larger divisor: $N = 7b + 3$.
2. Generate values of $N$ by incrementing $b \ge 0$: $b = 0 \implies 3$ $b = 1 \implies 10$ $b = 2 \implies 17$
3. Test each value against the second condition (remainder 2 when divided by 5):
   • $3 / 5 \implies \text{rem } 3$
   • $10 / 5 \implies \text{rem } 0$
   • $17 / 5 \implies 3 \text{ with rem } 2$ (Match found!)
4. The first common integer is 17. The pattern repeats every $\text{LCM}(5, 7) = 35$: $N = 35k + 17 \quad (k \in \mathbb{Z})$ Possible positive values: ${17, 52, 87, 122, \dots}$.

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Parity Algebra: Even and Odd Properties

Parity refers to the classification of an integer as either even ($2k$) or odd ($2k + 1$).

Strategic Parity Deductions

• If $x + y$ is odd, then one variable is even and the other is odd.
• If $x + y$ is even, then $x$ and $y$ share the same parity (both even or both odd).
• If $xy$ is odd, then $x$ and $y$ must both be odd.
• If $xy$ is even, then at least one variable must be even.
• The sum of an odd count of odd numbers is always odd ($1 + 3 + 5 = 9$).
• The sum of an even count of odd numbers is always even ($1 + 3 + 5 + 7 = 16$).

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Positive and Negative Sign Analysis

Tracking signs through inequalities and products is essential for GMAT algebra.

Product and Quotient Rules

1. $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$
2. $(+) \times (-) = (-)$
3. $\frac{(+)}{(+)} = (+)$ and $\frac{(-)}{(-)} = (+)$
4. $\frac{(+)}{(-)} = (-)$ and $\frac{(-)}{(+)} = (-)$

General Multi-Term Product Rule

For any product of non-zero real numbers:

• If the number of negative factors is even, the product is positive.
• If the number of negative factors is odd, the product is negative.

Powers and Inequalities with Signs

• For any real number $x \neq 0$, $x^2 > 0$, $x^4 > 0$, and generally $x^{\text{even}} > 0$.
• An even power destroys sign information: if $x^2 = 25$, $x$ could be $+5$ or $-5$.
• An odd power preserves sign information: if $x^3 = -64$, $x$ must be $-4$.
• If $a b > 0$, then $a$ and $b$ have the same sign (both positive or both negative).
• If $a b < 0$, then $a$ and $b$ have opposite signs.

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Worked Problem Solving Examples

Example 1: Large Modular Exponent Remainder

Problem: What is the remainder when $13^{45} + 17^{45}$ is divided by 15?

Solution:

1. Evaluate the base remainders modulo 15: $13 \equiv -2 \pmod{15} \quad (\text{since } 13 - 15 = -2)$ $17 \equiv 2 \pmod{15} \quad (\text{since } 17 - 15 = 2)$
2. Substitute the modular equivalents into the exponential expression: $13^{45} + 17^{45} \equiv (-2)^{45} + 2^{45} \pmod{15}$
3. Because 45 is an odd exponent, $(-2)^{45} = -(2^{45})$: $-(2^{45}) + 2^{45} = 0$
4. The sum is an exact multiple of 15, so the remainder is 0.

Example 2: Parity and Prime Equation

Problem: If $p, q,$ and $r$ are prime numbers such that $p + q + r = 40$ and $p < q < r$, what is the value of $p$?

Solution:

1. Analyze the parity of the equation: $\text{Prime}_1 + \text{Prime}_2 + \text{Prime}_3 = 40$ (even).
2. If all three primes were odd, their sum would be $\text{Odd} + \text{Odd} + \text{Odd} = \text{Odd}$. But 40 is even, so this is impossible.
3. To obtain an even sum from three integers, either all three must be even, or exactly one must be even and two must be odd: $\text{Even} + \text{Odd} + \text{Odd} = \text{Even}$
4. There is only one even prime number in mathematics: 2. Therefore, exactly one of the primes must be 2.
5. Since $p < q < r$, the smallest prime $p$ must be 2. (Verification: If $p = 2$, then $q + r = 38$. Primes summing to 38 include $q = 7$ and $r = 31$, satisfying $2 < 7 < 31$.)

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High-Frequency GMAT Traps

• Trap 1: The Zero Parity Blindspot: Neglecting that 0 is an even integer when evaluating algebraic constraints.
• Trap 2: Assuming Remainders Can Be Negative: If an algebraic simplification yields $-3$ modulo 8, the true positive remainder is $-3 + 8 = 5$.
• Trap 3: Dividing an Inequality by a Variable with Unknown Sign: If you divide or multiply an inequality by $x$ without knowing whether $x$ is positive or negative, you do not know whether to flip the inequality sign. Always check signs or multiply by $x^2$.