GMAT Quantitative: Number Properties and Arithmetic

Number properties and arithmetic are the bedrock of GMAT quantitative problem-solving. While the test doesn't require advanced calculus, its difficulty stems from how it twists fundamental concepts into complex logical puzzles. Mastering these foundational rules transforms your approach, allowing you to dissect problems efficiently, avoid common traps, and unlock solutions through reasoning rather than brute calculation.

Foundational Integer Classifications and Properties

At the heart of GMAT number theory lies the integer: numbers with no fractional or decimal part (..., -2, -1, 0, 1, 2, ...). Understanding how integers are categorized is your first strategic tool. Even integers are divisible by 2 (e.g., -4, 0, 6), while odd integers are not (e.g., -3, 1, 17). The test frequently exploits the basic arithmetic outcomes of these categories: even ± even = even, odd ± odd = even, and even ± odd = odd. Multiplication follows different rules: any integer multiplied by an even number yields an even result.

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, and 13. Crucially, 2 is the only even prime number. Every other even number is divisible by 2 and thus composite (having more than two divisors). A firm grasp of primes up to 50 is essential for factorization, a core skill for solving many GMAT problems. Positive integers greater than 1 that are not prime are composite numbers (e.g., 4, 6, 8, 9).

Divisibility, Factors, and Multiples

Divisibility rules are shortcuts that save invaluable time. For the GMAT, know these cold: A number is divisible by 2 if its last digit is even. By 3 if the sum of its digits is divisible by 3. By 4 if its last two digits form a number divisible by 4. By 5 if it ends in 0 or 5. By 6 if it is divisible by both 2 and 3. By 9 if the sum of its digits is divisible by 9. By 10 if it ends in 0.

These rules help you work with factors (divisors) and multiples. The greatest common divisor (GCD) or greatest common factor (GCF) of two numbers is the largest integer that divides both. The least common multiple (LCM) is the smallest positive integer that is a multiple of both. Finding these is critical for solving problems about repeating cycles, ratios, or combined events. The prime factorization method is the most reliable: for the GCD, take the lowest power of each common prime factor; for the LCM, take the highest power of all prime factors present.

Example: Find the GCD and LCM of 24 and 36.

• Prime factorize: $24=2^3\ast 3^1$, $36=2^2\ast 3^2$.
• GCD = $2^{min(3,2)}\ast 3^{min(1,2)}=2^2\ast 3^1=12$.
• LCM = $2^{max(3,2)}\ast 3^{max(1,2)}=2^3\ast 3^2=72$.

Arithmetic Operations, Absolute Value, and Order

Fluency with positive and negative number operations is non-negotiable. Remember: multiplying/dividing two numbers with the same sign yields a positive result; with opposite signs, a negative result. Adding a negative is the same as subtraction.

The absolute value of a number, denoted $|x|$, is its distance from zero on the number line, always a non-negative value. Thus, $|5|=5$ and $|-5|=5$. Equations and inequalities involving absolute value, like $|x-3|=7$, test your understanding that the expression inside can be either positive or negative (i.e., $x-3=7$ or $x-3=-7$).

The order of operations dictates the sequence for calculations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right), often abbreviated as PEMDAS. A classic GMAT trap is to present an expression like $8/2(2+2)$ to see if you mistakenly multiply before dividing.

Fractions, Decimals, and Remainders

Fractions represent a part of a whole, $\frac{numerator}{denominator}$. Key operations include simplifying by canceling common factors, finding common denominators for addition/subtraction, and for multiplication (multiply numerators and denominators directly) and division (multiply by the reciprocal). Comparing fractions is often easiest by finding a common denominator or cross-multiplying.

The concept of remainders is central to many advanced integer property questions. When an integer $n$ is divided by a positive integer $d$, we get $n=qd+r$, where $q$ is the quotient and $r$ is the remainder such that $0\le r<d$. GMAT problems often ask about patterns in remainders or the properties of $r$. For example, if $n$ leaves a remainder of 4 when divided by 7, it can be expressed as $n=7k+4$ for some integer $k$.

Common Pitfalls

1. Assuming numbers are positive. The GMAT routinely tests negative numbers and zero. Always consider if a variable could be zero, negative, or positive unless explicitly stated otherwise. A question asking "Is $x>y$?" becomes far trickier if $x$ and $y$ could be negative.
2. Misapplying divisibility rules. A common error is applying the rule for 3 to the number 3 itself in a convoluted way. Remember, the rule states: If the sum of the digits is divisible by 3, then the number is divisible by 3. It is a conditional test, not a definition.
3. Confusing factors and multiples. The factors of a number are less than or equal to it (except the number itself). The multiples of a number are greater than or equal to it. If the problem asks, "How many numbers less than 50 are multiples of 7?" you list 7, 14, 21, 28, 35, 42, 49. If it asks for factors of 50, you list 1, 2, 5, 10, 25, 50.
4. Overlooking the uniqueness of prime factorization. The Fundamental Theorem of Arithmetic states every integer greater than 1 can be uniquely represented as a product of prime numbers, ignoring order. This is a powerful tool for solving GCD/LCM problems and questions about the number of factors an integer has.

Summary

• Classify numbers strategically: Use the properties of integers (even/odd, positive/negative, prime/composite) as logical constraints to eliminate answer choices and reason through data sufficiency questions.
• Master the core tools: Divisibility rules, prime factorization, and the formulas for GCD and LCM are not just rote memory but essential algorithms for efficient problem-solving.
• Respect the order of operations and absolute value: These are frequent sources of "trap" answers. Always pause to interpret absolute value expressions correctly and methodically apply PEMDAS.
• Think in terms of remainders: The equation $n=qd+r$ is a versatile algebraic representation for a wide range of integer-based problems.
• Consider all number domains: Never assume a variable represents a positive integer unless the problem explicitly restricts it. Zero and negative numbers are often the key to the trickiest questions.