5.3 Factors, Multiples, Prime Factorization & Divisibility Rules

5.3 Factors, Multiples, Prime Factorization & Divisibility Rules

Number theory concepts—including factors, multiples, prime factorization, and divisibility rules—form the hidden backbone of mathematical efficiency on the Wonderlic Basic Skills Test (WBST). When solving fraction reductions, common denominator calculations, and word problems, candidates who calculate by trial-and-error long division bleed away valuable time.

With an average of 26.67 seconds per quantitative item, you must be able to decompose numbers rapidly into their foundational prime components and apply universal divisibility tests on sight. These techniques allow you to bypass heavy arithmetic entirely, verify answers in under 5 seconds, and systematically eliminate plausible distractors.

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Prime Numbers vs. Composite Numbers: Definitions and Boundary Traps

Every whole number greater than 1 falls into one of two fundamental categories:

• Prime Number: A positive integer strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself.
• Composite Number: A positive integer strictly greater than 1 that has more than two distinct positive divisors (meaning it can be formed by multiplying two smaller positive integers).

                      THE WHOLE NUMBER TAXONOMY (n ≥ 0)
                                      |
         +----------------------------+----------------------------+
         |                                                         |
  [ Special Boundary: 0 & 1 ]                              [ Integers n ≥ 2 ]
  * 0 is neither prime nor composite                               |
  * 1 is NEITHER prime nor composite              +----------------+----------------+
    (has only ONE distinct factor: 1)             |                                 |
                                          [ Prime Numbers ]                [ Composite Numbers ]
                                          Exactly 2 factors                3 or more factors
                                          (e.g., 2, 3, 5, 7, 11)           (e.g., 4, 6, 8, 9, 12)

Critical Boundary Traps

1. The Number 1: The number 1 is NEITHER prime nor composite. It fails the definition of prime because it does not possess two distinct factors (its only factor is 1).
2. The Number 2: The number 2 is the smallest prime number and the ONLY even prime number. Every other even number is divisible by 2 and is therefore composite.
3. The Primes Under 50: You should have the first 15 prime numbers memorized for instant recognition on test day: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$

High-Frequency "False Prime" Traps on the WBST

Test writers deliberately include composite numbers that look prime because they are odd and do not end in 5. Memorize these frequent traps:

• 51 is composite: $5 + 1 = 6$ (divisible by 3; $3 × 17 = 51$).
• 57 is composite: $5 + 7 = 12$ (divisible by 3; $3 × 19 = 57$).
• 87 is composite: $8 + 7 = 15$ (divisible by 3; $3 × 29 = 87$).
• 91 is composite: Divisible by 7 and 13 ($7 × 13 = 91$).
• 119 is composite: Divisible by 7 and 17 ($7 × 17 = 119$).

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Prime Factorization & Factor Trees

The Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers in a way that is unique up to the order of the factors. This prime decomposition is the "DNA" of the number.

Constructing a Factor Tree Rapidly

To find the prime factorization of a composite number on scratch paper:

1. Choose any two obvious factors of the number (start with 2 if even, or 5/10 if ending in 0 or 5).
2. Branch the number into those two factors.
3. If a factor is prime, circle it. If it is composite, branch it again.
4. Continue until all leaf nodes are circled prime numbers.
5. Collect identical primes into exponential prime-power notation.

                         FACTOR TREE FOR 360
                                360
                               /   \
                             36  ×  10
                            /  \    / \
                           6 × 6   (2) (5)
                          / \ / \
                        (2)(3)(2)(3)
                        
   Collected Prime Factors: 2 × 2 × 2 × 3 × 3 × 5
   Canonical Prime-Power Notation: 360 = 2³ × 3² × 5¹

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

Understanding GCF and LCM is essential for fraction operations and applied word problems on the WBST.

• Greatest Common Factor (GCF / GCD): The largest positive integer that divides evenly into two or more given numbers without leaving a remainder. GCF is used to reduce fractions to their simplest terms and to divide quantities into equal groups.
• Least Common Multiple (LCM): The smallest positive integer that is an exact multiple of two or more given numbers. LCM is used to find the Least Common Denominator (LCD) when adding or subtracting fractions and to calculate repeating cycle intervals.

The Prime Factorization Method for GCF and LCM

Once you express two numbers in their prime-power forms, you can find both GCF and LCM instantly using a simple rule:

• For GCF: Take the LOWEST power of each COMMON prime factor.
• For LCM: Take the HIGHEST power of ALL prime factors present in either number.

Comprehensive Worked Comparison: Numbers 72 and 120

First, find the prime factorizations: $72 = 8 × 9 = 2^3 × 3^2$ $120 = 10 × 12 = (2 × 5) × (2^2 × 3) = 2^3 × 3^1 × 5^1$

The "Ladder" (Upside-Down Division) Shortcut

When scratch paper space and time are minimal, the ladder method computes GCF and LCM simultaneously:

1. Write the numbers side by side: | 72 120
2. Divide both by a shared common factor (e.g., 2): | 36 60
3. Divide again by 2: | 18 30
4. Divide again by 2: | 9 15
5. Divide by 3: | 3 5 (Stop: 3 and 5 are coprime).

• GCF = Product of the vertical left column numbers: $2 × 2 × 2 × 3 = 24$.
• LCM = Product of the vertical column AND the bottom row (in an "L" shape): $24 × 3 × 5 = 360$.

Word Problem Scenario Recognition: GCF vs. LCM

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Comprehensive Divisibility Rules: The Speed Master Reference

Divisibility tests allow you to determine whether an integer is divisible by another number without performing long division. On the WBST, these rules save up to 20 seconds per item.

Advanced Shortcut: Divisibility by 8 Without 3-Digit Division

Dividing a 3-digit number by 8 in your head can be slow. Use the Hundreds Parity Shortcut:

• If the hundreds digit is EVEN, look only at the last two digits: if they are divisible by 8, the whole number is divisible by 8.
  • In 624: Hundreds digit 6 is even. Check 24: $24 ÷ 8 = 3$. Divisible!
• If the hundreds digit is ODD, add 4 to the last two digits: if the sum is divisible by 8, the whole number is divisible by 8.
  • In 728: Hundreds digit 7 is odd. Add 4 to last two digits: $28 + 4 = 32$. Since $32 ÷ 8 = 4$, 728 is divisible by 8!

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Strategic Application: Rapid Distractor Elimination & Computation Simplification

Under the 26.67-second pressure of the WBST Quantitative section, divisibility rules function as high-speed filters:

1. Fraction Reduction: If asked to reduce $\frac{162}{216}$:
   • Sum of digits for 162: $1 + 6 + 2 = 9$ (divisible by 9).
   • Sum of digits for 216: $2 + 1 + 6 = 9$ (divisible by 9).
   • Immediately divide numerator and denominator by 9: $\frac{162 ÷ 9}{216 ÷ 9} = \frac{18}{24}$.
   • Divide by 6: $\frac{18 ÷ 6}{24 ÷ 6} = \frac{3}{4}$. You reach lowest terms in 6 seconds without testing factors of 2 or 3 repeatedly.
2. Instant Multiple-Choice Elimination: If a question asks which number is divisible by 6, immediately eliminate any odd answer choice in 1 second (violates rule for 2). Then test the digit sums of the remaining even choices (rule for 3).
3. Prime Number Identification: When presented with four multi-digit numbers and asked to identify the prime number, eliminate options ending in even digits or 5 first. Then calculate digit sums to eliminate multiples of 3. In over 80% of WBST questions, this leaves exactly one candidate standing within 8 seconds.