2.1 Integers, Divisibility & Prime Factorization

2.1 Integers, Divisibility & Prime Factorization

Number properties form the foundational bedrock of GRE Quantitative Reasoning. Questions testing integer properties, divisibility, prime numbers, and remainders appear frequently across both Multiple-Choice and Quantitative Comparison (QC) formats. Mastering these rules enables you to solve complex problems rapidly without relying on tedious brute-force calculations.

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Integer Definitions & Core Properties

An integer is any whole number in the set $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Integers include positive whole numbers, negative whole numbers, and zero.

GRE Quantitative Comparison Trap: Never assume a variable represents an integer unless the problem explicitly states that $x$ or $y$ is an integer. If a question states "$x > 0$," $x$ could be a fraction like $\frac{1}{2}$ or an irrational number like $\sqrt{2}$.

Essential Subsets of Real Numbers

• Positive Integers: $\{1, 2, 3, 4, \dots\}$ (also called counting numbers or natural numbers).
• Non-Negative Integers: $\{0, 1, 2, 3, 4, \dots\}$ (includes zero).
• Zero ($0$): Zero is an even integer. It is neither positive nor negative. Any number multiplied by zero is zero ($n \times 0 = 0$), and division by zero is undefined.

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Divisibility Rules (2 through 10)

An integer $a$ is divisible by an integer $b$ (where $b \neq 0$) if there exists an integer $q$ such that $a = b \times q$, leaving a remainder of $0$. Below is the definitive reference table for divisibility test shortcuts required on the GRE:

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Prime Numbers & Prime Factorization

A prime number is a positive integer greater than $1$ that has exactly two distinct positive divisors: $1$ and itself. A positive integer greater than $1$ that is not prime is a composite number.

Crucial Prime Number Facts

• $1$ is NEITHER prime nor composite. It has only one positive divisor ($1$).
• $2$ is the ONLY even prime number. Every other even number is divisible by $2$ and thus composite.
• $2$ is the smallest prime number.
• Primes under 50 (Memorize these 15 numbers): $$\mathbf{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}$$

Prime Factorization Trees

Every composite number can be uniquely expressed as a product of prime numbers (Fundamental Theorem of Arithmetic). To build a prime factorization tree, break the number down into any two factors and continue breaking non-prime factors down until only primes remain.

Worked Example: Prime Factorization of 360

$$360 = 36 \times 10 = (6 \times 6) \times (2 \times 5) = (2 \times 3) \times (2 \times 3) \times 2 \times 5 = 2^3 \times 3^2 \times 5^1$$

Formula for Total Number of Positive Factors

If the prime factorization of a positive integer $N$ is:

$$N = p_1^{a} \times p_2^{b} \times p_3^{c} \dots$$

where $p_1, p_2, p_3$ are distinct prime factors, then the total number of positive factors of $N$ is given by:

$$\text{Total Factors} = (a + 1)(b + 1)(c + 1)\dots$$

Example: For $360 = 2^3 \times 3^2 \times 5^1$, the number of factors is $(3+1)(2+1)(1+1) = 4 \times 3 \times 2 = 24$ factors.

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Greatest Common Divisor (GCD) & Least Common Multiple (LCM)

• Greatest Common Divisor (GCD): The largest positive integer that divides two or more integers without a remainder (also called Highest Common Factor, HCF).
• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more integers.

Prime Factorization Method for GCD and LCM

1. Express each number in its prime factorization.
2. For GCD: Multiply the lowest power of each common prime factor.
3. For LCM: Multiply the highest power of every prime factor present in any of the numbers.

Worked Example: Finding GCD and LCM of 48 and 180

• Prime Factorization of $48$: $48 = 16 \times 3 = 2^4 \times 3^1$
• Prime Factorization of $180$: $180 = 18 \times 10 = 2^2 \times 3^2 \times 5^1$

$$\text{GCD}(48, 180) = 2^{\min(4,2)} \times 3^{\min(1,2)} = 2^2 \times 3^1 = 4 \times 3 = 12$$ $$\text{LCM}(48, 180) = 2^{\max(4,2)} \times 3^{\max(1,2)} \times 5^{\max(0,1)} = 2^4 \times 3^2 \times 5^1 = 16 \times 9 \times 5 = 720$$

Key Formula: For any two positive integers $a$ and $b$: $\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b$ Verification: $12 \times 720 = 8,640$, and $48 \times 180 = 8,640$.

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Even & Odd Parity Rules

Parity refers to whether an integer is even or odd. Remember: Zero ($0$) is even, negative numbers can be even or odd (e.g., $-4$ is even, $-7$ is odd).

Arithmetic Operations with Parity

GRE Strategy Tip: In Quantitative Comparison questions involving parity, test $0$ as an even number. If an expression involves products, remember that a single even factor makes the entire product even!

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Remainder Theory

When an integer $N$ (the dividend) is divided by a positive integer $d$ (the divisor), it yields an integer $q$ (the quotient) and a non-negative integer $r$ (the remainder):

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

Key Remainder Properties

1. Range Restriction: The remainder $r$ must always be greater than or equal to $0$ and strictly less than the divisor $d$.
2. Modular Arithmetic Rules:
   • $(a + b) \pmod d = ((a \pmod d) + (b \pmod d)) \pmod d$
   • $(a \times b) \pmod d = ((a \pmod d) \times (b \pmod d)) \pmod d$
3. Cyclicity: Powers of numbers produce repeating patterns of remainder cycles when divided by a fixed base. For example, powers of $7$ mod $5$ follow a cycle of length 4: $7^1 \equiv 2, 7^2 \equiv 4, 7^3 \equiv 3, 7^4 \equiv 1$ (mod 5).