GRE Arithmetic Number Properties and Divisibility

Mastering number properties and divisibility is non-negotiable for a high GRE quantitative score. These concepts are the bedrock upon which countless arithmetic, algebra, and word problems are built, appearing frequently in both Quantitative Comparison and Problem Solving questions. Your ability to swiftly manipulate integers, recognize patterns, and apply foundational rules directly translates to efficiency and accuracy under time pressure.

Foundations: Integers, Primes, Factors, and Multiples

Every GRE number property question deals with integers, which are whole numbers including zero and negatives. Within this set, understanding a few key definitions is crucial. A factor (or divisor) of an integer $n$ is an integer that divides $n$ with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Conversely, a multiple of an integer $n$ is the product of $n$ and any integer. The multiples of 3 are 3, 6, 9, 12, and so on. A prime number is an integer greater than 1 that has exactly two distinct positive factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, and 13. Memorizing primes up to 50 is a valuable time-saver.

On the GRE, you must often distinguish between these concepts. A Quantitative Comparison question might ask:

Quantity A: The number of prime factors of 36.

Quantity B: The number of distinct prime factors of 36.

The trap is confusing "prime factors" with "distinct prime factors." The prime factorization of 36 is $2\times 2\times 3\times 3$, so it has four prime factors but only two distinct ones (2 and 3). Recognizing this precise terminology is a tested skill.

Parity Rules: The Behavior of Even and Odd Integers

Even integers are divisible by 2 (e.g., -4, 0, 2, 16), while odd integers are not (e.g., -3, 1, 7, 15). The rules governing their sums, differences, and products are simple but powerful tools for logical deduction without calculation.

• Addition/Subtraction: even ± even = even; odd ± odd = even; even ± odd = odd.
• Multiplication: even × any integer = even; odd × odd = odd.

Consider this GRE-style scenario: If $a$ and $b$ are consecutive integers, is $a^2+b^2$ odd or even? Since consecutive integers have opposite parity, one is even and one is odd. An even squared is even, an odd squared is odd. According to the addition rule, even + odd = odd. Therefore, $a^2+b^2$ is always odd. This kind of reasoning is faster and more reliable than plugging in numbers.

Divisibility Tests and Shortcut Rules

Divisibility rules allow you to determine if one integer divides another without performing long division. For the GRE, the rules for 2, 3, 4, 5, 6, 9, and 10 are most common.

• 2: Last digit is even (0, 2, 4, 6, 8).
• 3: Sum of digits is divisible by 3.
• 4: Last two digits form a number divisible by 4.
• 5: Last digit is 0 or 5.
• 6: Divisible by both 2 and 3.
• 9: Sum of digits is divisible by 9.
• 10: Last digit is 0.

A strategic application is simplifying problems. For instance, a question asking for the remainder when a large number like 7,258 is divided by 9 can be solved by summing digits: 7+2+5+8=22. Since 22 divided by 9 leaves a remainder of 4, the original number also leaves a remainder of 4. Always check the applicability of these rules before embarking on more laborious calculations.

The Fundamental Theorem of Arithmetic and Prime Factorization

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. This prime factorization is the most powerful tool for solving complex number property questions.

Finding the prime factorization involves repeated division by primes. For example: $$180=2\times 90=2\times 2\times 45=2^2\times 3\times 15=2^2\times 3^2\times 5$$

This factorization allows you to quickly list all factors, understand divisibility, and, most importantly, compute the greatest common factor (GCF) and least common multiple (LCM). The GCF of two numbers is the largest factor that divides both. The LCM is the smallest positive multiple that both numbers share.

• To find the GCF using prime factorization: Take the product of the lowest powers of each prime common to all numbers.
• To find the LCM using prime factorization: Take the product of the highest powers of all primes present in any of the numbers.

For numbers 24 and 60:

• $24=2^3\times 3$
• $60=2^2\times 3\times 5$
• GCF = $2^2\times 3=12$ (lowest powers of common primes: $2^2$ and $3^1$).
• LCM = $2^3\times 3\times 5=120$ (highest powers of all primes: $2^3$, $3^1$, $5^1$).

Applications: GCF and LCM in GRE Problem Solving

GCF and LCM concepts frequently underpin word problems. The GCF is key to problems about dividing groups into smaller, equal subsets ("greatest number of identical packages"). The LCM is key to problems about repeating events or cycles ("when will two trains leave the station together again?").

A classic GRE trap involves confusing the two. Consider: "Two lights blink every 6 and 8 seconds, respectively. They blink together at noon. How many times will they blink together in the next 90 seconds?" The cycle time is the LCM of 6 and 8, which is 24 seconds. They blink together at 0, 24, 48, 72, and 96 seconds. Within 90 seconds, this happens at 0, 24, 48, and 72 seconds—four times. A common error is to use the GCF (2) instead of the LCM, leading to an incorrect count. Always ask yourself: Am I looking for a divisor of the numbers (GCF) or a multiple of them (LCM)?

In Quantitative Comparison, you might be asked to compare the GCF and LCM of two numbers. Remember that for any two positive integers, LCM(a, b) ≥ GCF(a, b), and they are equal only if the two numbers are identical.

Common Pitfalls

Pitfall 1: Confusing factors and multiples. A factor is a "part" of a number (smaller or equal), while a multiple is a "product" of the number (larger or equal). In stress, test-takers often reverse these. Correction: Use the mnemonic "Few Factors, Many Multiples." The list of factors is finite, while multiples go on forever.

Pitfall 2: Overlooking 1 and the number itself. When listing factors, it's easy to forget that 1 and the number itself are always factors. When testing for primality, remember that a prime number has exactly two distinct positive factors, which excludes 1 (which has only one factor).

Pitfall 3: Misapplying rules to zero and negatives. Zero is an even integer, but it is neither positive nor negative. Divisibility by zero is undefined. While parity rules hold for negatives, some test-takers hesitate. Correction: Remember that the definitions of even and odd are based on divisibility by 2, which applies cleanly to all integers.

Pitfall 4: Assuming uniqueness in non-prime factorizations. Only prime factorizations are guaranteed unique by the Fundamental Theorem. For example, 12 can be factored as 3×4 or 2×6, but only $2^2\times 3$ is the unique prime factorization. Rely on prime factorization for GCF/LCM and definitive conclusions.

Summary

• Prime factorization is your master key. Use the Fundamental Theorem of Arithmetic to break any integer into its unique prime components. This process unlocks solutions for factors, multiples, GCF, and LCM problems.
• Parity and divisibility rules are for logic, not just calculation. Use the properties of even/odd numbers and quick divisibility tests to eliminate answer choices and deduce results without extensive computation.
• Distinguish GCF from LCM applications. GCF answers "greatest number that divides" questions (like splitting into groups), while LCM answers "when will events coincide" questions (like repeating cycles).
• Precision in terminology matters on the GRE. Know the exact difference between "factor" and "multiple," "prime factor" and "distinct prime factor," and "integer" and "positive integer."
• Practice recognizing common traps. Be vigilant about the number 1, the behavior of zero, and the temptation to confuse related concepts like GCF and LCM in word problems.