2.1 Integers, Divisibility Rules, Prime Factorization, and LCM/GCD

2.1 Integers, Divisibility Rules, Prime Factorization, and LCM/GCD

Quick Summary: On the GMAT Focus Edition Quantitative Reasoning section, calculators are strictly prohibited. Mastery of integer classifications, mental divisibility rules (2 through 12), canonical prime factorization trees, and the algebraic relationship $\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b$ allows you to dismantle complex Problem Solving questions rapidly without laborious manual arithmetic.

Arithmetic and number properties form the computational and conceptual backbone of the GMAT Focus Quantitative Reasoning measure. Because all 21 Quantitative Reasoning questions are Problem Solving items delivered without calculator access, the exam tests your structural understanding of how numbers interact rather than your ability to perform tedious pencil-and-paper calculations.

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Integer Classification and the Special Properties of Zero

The set of integers (denoted mathematically as $\mathbb{Z}$) consists of all whole numbers, their negative opposites, and zero:

$\mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}$

Notice that fractions, non-terminating decimals, irrational numbers (such as $\sqrt{2}$ or $\pi$), and imaginary numbers are not integers. On the GMAT, questions frequently impose integer constraints using specific terms that you must interpret with complete precision:

• Natural Numbers (Counting Numbers): The positive integers ${1, 2, 3, 4, \dots}$.
• Whole Numbers: Non-negative integers ${0, 1, 2, 3, \dots}$.
• Non-Negative Integers: All integers greater than or equal to zero ($x \ge 0$). This includes 0!
• Non-Positive Integers: All integers less than or equal to zero ($x \le 0$). This includes 0!

The Critical Arithmetic Rules of Zero

Zero is a frequent source of test-day traps because it possesses unique mathematical properties that separate it from all other integers:

1. Zero is an Integer: Zero has no fractional or decimal component.
2. Zero is Even: By formal mathematical definition, an integer $n$ is even if and only if $n = 2k$ for some integer $k$. Setting $k = 0$ yields $0 = 2(0)$. Because 0 divides by 2 with a remainder of 0, zero is strictly even.
3. Zero is Neither Positive nor Negative: Zero is the neutral origin on the real number line. It divides positive values from negative values. If a problem states that $n$ is a "positive integer," $n$ cannot be 0 ($n \ge 1$).
4. Zero is a Multiple of Every Integer: For any integer $k$, $0 = k \times 0$. Thus, 0 is a multiple of 3, a multiple of 7, and a multiple of 84. However, zero is never a factor or divisor of any number.
5. Division by Zero is Undefined: While $0 / k = 0$ for any non-zero integer $k$, the expression $k / 0$ is mathematically undefined. In algebraic expressions such as $\frac{x - 4}{y + 2}$, you must immediately recognize the implied domain restriction $y \neq -2$.

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Mental Divisibility Rules (2 through 12)

An integer $N$ is divisible by a non-zero integer $d$ if the quotient $N / d$ is an integer with a remainder of 0. In such cases, $d$ is a factor (or divisor) of $N$, and $N$ is a multiple of $d$.

To solve Problem Solving questions in under two minutes without a calculator, you must execute mental divisibility checks automatically.

The Coprime Factor Rule for Composite Divisors

A common mistake on composite divisibility is testing non-coprime factors. To test whether an integer is divisible by a composite divisor $C = a \times b$, the factors $a$ and $b$ must be relatively prime (meaning $\text{GCD}(a, b) = 1$):

• To test for divisibility by 12, you must test 3 and 4 ($ ext{GCD}(3, 4) = 1$). You cannot test 2 and 6, because a number such as 18 is divisible by both 2 and 6, yet it is not divisible by 12 ($18 / 12 = 1.5$).
• To test for divisibility by 18, test 2 and 9 ($ ext{GCD}(2, 9) = 1$).
• To test for divisibility by 36, test 4 and 9 ($ ext{GCD}(4, 9) = 1$).

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Prime Numbers and Canonical Prime Factorization

A prime number is an integer greater than 1 that possesses exactly two distinct positive divisors: 1 and itself.

Fundamental Prime Facts to Memorize

• 1 is NOT a prime number: The number 1 has only one positive divisor (itself).
• 2 is the ONLY even prime number: Every other even integer is divisible by 2 and therefore composite. The number 2 is also the smallest prime number.
• All primes greater than 2 are odd, but not all odd numbers are prime (e.g., 9, 15, 21, 25, 27 are composite).
• The first 15 prime numbers: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer $N > 1$ can be expressed as a product of prime numbers in exactly one way, up to the order of the factors. This canonical prime factorization takes the form:

$N = p_1^{a} \cdot p_2^{b} \cdot p_3^{c} \cdots$

where $p_1, p_2, p_3$ are distinct prime numbers and $a, b, c$ are positive integer exponents.

Formula for the Total Number of Positive Divisors

Given the canonical prime factorization $N = p_1^{a} \cdot p_2^{b} \cdot p_3^{c} \dots$, any positive factor $d$ of $N$ must have the prime factorization:

$d = p_1^{x} \cdot p_2^{y} \cdot p_3^{z} \dots$

where $0 \le x \le a$, $0 \le y \le b$, and $0 \le z \le c$. By the Fundamental Counting Principle, the number of choices for exponent $x$ is $(a + 1)$, for $y$ is $(b + 1)$, and for $z$ is $(c + 1)$. Therefore:

$\text{Total Positive Divisors} = (a + 1)(b + 1)(c + 1)\cdots$

Example: Total Divisors of 360

Prime factorizing 360: $360 = 36 \times 10 = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5^1$ Applying the formula: $\text{Total Divisors} = (3 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 2 = 24$

Exam Takeaway on Perfect Squares: An integer has an odd number of total positive factors if and only if it is a perfect square. In any perfect square, all prime exponents $a, b, c$ are even numbers, meaning $(a + 1), (b + 1), (c + 1)$ are all odd numbers, producing an odd product.

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

Greatest Common Divisor (GCD)

The Greatest Common Divisor (or Greatest Common Factor, GCF) of two or more positive integers is the largest positive integer that divides all of them without leaving a remainder.

Method: Prime factorize each number and take the lowest exponent of each prime factor that is common to all factorizations.

Least Common Multiple (LCM)

The Least Common Multiple of two or more positive integers is the smallest positive integer that is a multiple of all of them.

Method: Prime factorize each number and take the highest exponent of every prime factor present in any of the factorizations.

The Core Product Identity

For any two positive integers $a$ and $b$:

$\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b$

Caution: This product formula applies strictly to two integers. For three integers, $\text{GCD}(a, b, c) \times \text{LCM}(a, b, c) \neq a \times b \times c$ in general.

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Worked Problem Solving Examples

Example 1: Finding Constrained Divisors

Problem: If $N = 2^4 \times 3^3 \times 5^2 \times 7^1$, how many positive divisors of $N$ are multiples of 36?

Solution:

1. Express the constraint condition (being a multiple of 36) in prime factored form: $36 = 4 \times 9 = 2^2 \times 3^2$
2. Factor $2^2 \times 3^2$ out of the prime factorization of $N$: $N = (2^2 \times 3^2) \times \left(2^{4-2} \times 3^{3-2} \times 5^2 \times 7^1\right) = 36 \times \left(2^2 \times 3^1 \times 5^2 \times 7^1\right)$
3. Any divisor of the remaining quotient $Q = 2^2 \times 3^1 \times 5^2 \times 7^1$, when multiplied by 36, produces a unique divisor of $N$ that is a multiple of 36.
4. Apply the divisor formula to $Q$: $\text{Number of factors} = (2 + 1)(1 + 1)(2 + 1)(1 + 1) = 3 \times 2 \times 3 \times 2 = 36$ There are 36 positive divisors of $N$ that are multiples of 36.

Example 2: Reconstructing an Integer from GCD and LCM

Problem: The greatest common divisor of two positive integers $x$ and 108 is 18, and their least common multiple is 540. What is the value of $x$?

Solution:

1. Use the two-integer identity: $x \times 108 = \text{GCD}(x, 108) \times \text{LCM}(x, 108)$.
2. Substitute the given values: $x \times 108 = 18 \times 540$
3. Isolate $x$ using mental factorization rather than large multiplication: $x = \frac{18 \times 540}{108} = \frac{18}{108} \times 540 = \frac{1}{6} \times 540 = 90$
4. Check prime factorizations:
   • $108 = 2^2 \times 3^3$
   • $90 = 2^1 \times 3^2 \times 5^1$
   • $\text{GCD} = 2^{\min(2, 1)} \times 3^{\min(3, 2)} = 2^1 \times 3^2 = 18$
   • $\text{LCM} = 2^{\max(2, 1)} \times 3^{\max(3, 2)} \times 5^{\max(0, 1)} = 2^2 \times 3^3 \times 5^1 = 4 \times 27 \times 5 = 540$ Everything checks out; $x = 90$.

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High-Frequency GMAT Traps & Pacing Strategies

• Trap 1: "Divisors" vs. "Distinct Prime Factors": Read the question stem carefully. If an item asks for the number of prime factors of 72, the answer is 5 ($2 \times 2 \times 2 \times 3 \times 3$). If it asks for the number of distinct prime factors, the answer is 2 (only 2 and 3). If it asks for total positive divisors, the answer is $(3 + 1)(1 + 1) = 12$.
• Trap 2: Forgetting Negative Factors: Unless the prompt explicitly states "positive divisors" or "positive factors," factors can be negative. For example, 6 has positive factors ${1, 2, 3, 6}$ and total integer factors ${\pm 1, \pm 2, \pm 3, \pm 6}$ (8 total).
• Trap 3: Assuming 1 is Prime: 1 is neither prime nor composite. Including 1 in prime counts will invalidate your answer.