6.1 Number Properties & Arithmetic Operations

Number Properties & Arithmetic Operations

The Executive Assessment (EA) Quantitative section heavily tests your intuition and mastery of basic number properties. Rather than testing complex calculus or trigonometry, the EA focuses on deep understanding of integers, primes, absolute values, and exponents. This section will rebuild your foundational math skills and highlight common traps designed to test your attention to detail.

Integers and Number Classifications

Integers are whole numbers, including negative values, zero, and positive values (..., -3, -2, -1, 0, 1, 2, 3, ...). It is crucial to remember that 0 is an integer, and it is an even integer. However, 0 is neither positive nor negative.

When a problem specifies "positive integers," it means {1, 2, 3, ...}. If it specifies "non-negative integers," it means {0, 1, 2, 3, ...}.

Odd and Even Number Rules

Understanding how odd and even numbers interact is vital for the EA, especially for Data Sufficiency questions.

Addition and Subtraction:

• Even ± Even = Even
• Odd ± Odd = Even
• Even ± Odd = Odd

Multiplication:

• Even × Even = Even
• Odd × Odd = Odd
• Even × Odd = Even

Trap Alert: Division does not have guaranteed odd/even outcomes. For instance, an even divided by an even could be even (4/2 = 2), odd (6/2 = 3), or not an integer at all (4/6).

Positive and Negative Number Rules

Similar rules apply when multiplying and dividing positive and negative numbers:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

Prime Numbers and Factorization

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

Important notes on primes:

• 2 is the only even prime number.
• 1 is NOT a prime number because it only has one divisor (itself).
• All prime numbers greater than 2 are odd, but not all odd numbers are prime (e.g., 9, 15, 21, 27 are composite).

Prime Factorization is the process of breaking down a composite number into the product of its prime factors. For example, the prime factorization of 60 is: 60 = 6 × 10 = (2 × 3) × (2 × 5) = 2² × 3 × 5.

Greatest Common Factor (GCF) and Least Common Multiple (LCM)

• Greatest Common Factor (GCF): The largest integer that divides two or more numbers without leaving a remainder. To find it, use the lowest power of each common prime factor. Example: GCF of 12 (2² × 3) and 18 (2 × 3²) is 2¹ × 3¹ = 6.
• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers. To find it, use the highest power of all prime factors present. Example: LCM of 12 (2² × 3) and 18 (2 × 3²) is 2² × 3² = 4 × 9 = 36.

Divisibility Rules and Remainders

Knowing divisibility shortcuts can save you precious minutes on the exam:

• 2: The last digit is even (0, 2, 4, 6, 8).
• 3: The sum of the digits is divisible by 3.
• 4: The number formed by the last two digits is divisible by 4.
• 5: The last digit is 0 or 5.
• 6: The number is divisible by both 2 and 3.
• 8: The number formed by the last three digits is divisible by 8.
• 9: The sum of the digits is divisible by 9.
• 10: The last digit is 0.

Remainders: A remainder is what is left over when an integer cannot be divided evenly. For example, 17 divided by 5 yields a quotient of 3 and a remainder of 2. The formula is: Dividend = (Quotient × Divisor) + Remainder.

Absolute Value

The absolute value of a number, denoted as |x|, represents its distance from zero on the number line. Therefore, absolute value is always non-negative.

• |5| = 5
• |-5| = 5
• |0| = 0

When solving equations with absolute values, you must consider two cases: the positive case and the negative case. For example, if |x| = 5, then x could be 5 or -5.

Exponents and Radicals

An exponent indicates how many times a base is multiplied by itself. For example, 3⁴ = 3 × 3 × 3 × 3 = 81.

Key Exponent Rules:

Radicals (Square Roots)

A square root of a number n is a value that, when multiplied by itself, gives n. For example, √25 = 5. Note that the radical symbol (√) generally denotes the principal (positive) square root.

Properties of square roots:

• √(ab) = √a × √b
• √(a/b) = √a / √b

• Trap Alert: You cannot split addition or subtraction: √(a + b) ≠ √a + √b. For instance, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.

Mastering these foundational rules will give you the agility needed to tackle the Quantitative Reasoning section effectively. Practice these concepts until recognizing the prime factorization or exponent rule becomes second nature.