The Unbreakable Code Mastering Prime Factorization for GMAT Number Properties

The Unbreakable Code: Mastering Prime Factorization for GMAT Number Properties Every positive integer greater than 1 has a unique identity4its prime factorization.

This concept is the bedrock of advanced number theory on the GMAT, serving as the most powerful tool for understanding factors, multiples, divisibility, GCD, and LCM.

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Prime and Composite Numbers: The Building Blocks Prime Numbers

Composite Numbers

A positive integer greater than 1 with exactly two distinct positive

A positive integer greater than 1 with more than two distinct positive

factors: 1 and itself.

factors. These can be formed by multiplying smaller prime numbers.

Examples: 2, 3, 5, 7, 11, 13, 17, 19

Examples: 4 (factors: 1, 2, 4)

Note: 2 is the only even prime

6 (factors: 1, 2, 3, 6) 9 (factors: 1, 3, 9)

The Number 1 The number 1 is unique4it is neither prime nor composite. It has only one positive factor (itself).

Zero and Negatives By convention, prime and composite numbers are defined only for positive integers greater than 1.

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The Fundamental Theorem of Arithmetic Theorem: Every integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers, and this representation is unique, apart from the order of the prime factors.

This is one of the most important theorems in number theory, underpinning the entire concept of prime factorization. Every number has a unique "prime fingerprint."

Unique Identity

Building Blocks

Universal Truth

12 is always 2 × 2 × 3 (or 2² × 3), never 2 ×

Prime numbers are the fundamental

This theorem applies to every positive

6 or 3 × 4 as a final prime factorization

building blocks of all integers greater than

integer, making it a cornerstone of number

1

theory

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Division Method (Ladder Method) Start by dividing the number by the smallest prime number (2) that divides it evenly. Continue dividing the quotient by the smallest prime number until you reach a prime quotient.

Example: Prime factorize 180 2 | 180 2 | 90 3 | 45 3 | 15 5|5 |1 Result: 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5¹

01

02

Start with smallest prime

Continue dividing

Begin dividing by 2, the smallest prime number

Keep dividing by the smallest prime that works

03

04

Reach prime quotient

Express with exponents

Stop when you reach a prime number

Write the final answer using exponential notation

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Factor Tree Method Start with the number at the top, branch out into any two factors, and continue branching each composite factor until all branches end in prime numbers.

Example: Prime factorize 180 180 / \ 18

10

/\

/\

2 9 2 5 /\ 3 3

Collecting Prime Factors All prime numbers at the ends of branches: 2, 3, 3, 2, 5 Sorted with exponents: 2² × 3² × 5¹ Both methods yield the same unique prime factorization, confirming the Fundamental Theorem of Arithmetic.

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Application: Finding All Factors Once you have the prime factorization N = p¡_ × p¢_ × p£ , any factor of N will be in the form p¡ã × p¢º × p£¿ where 0 f x f a, 0 f y f b, 0 f z f c.

1

2

Prime Factorization

Factors of Each Prime

180 = 2² × 3² × 5¹

2²: {1, 2, 4} 3²: {1, 3, 9} 5¹: {1, 5}

3 Combine All Possibilities Take every combination: 1×1×1=1, 1×1×5=5, 2×3×5=30, 4×9×5=180, etc.

This systematic approach ensures you don't miss any factors4a crucial skill for GMAT problem-solving. www.goalisb.com | Shruti P | ISB, IIM, MBA Expert | contact@goalisb.com | https://www.youtube.com/@Goalisb

Counting the Number of Factors Formula: If N = p¡_ × p¢_ × p£ (where p¡, p¢, p£ are distinct primes), then the total number of positive factors is: (a + 1)(b + 1)(c + 1)...

Example: How many factors does 180 have?

180 = 2² × 3² × 5¹ Exponents: a=2, b=2, c=1 Number of factors = (2+1)(2+1)(1+1) = 3 × 3 × 2 = 18 factors

18

3

100%

Total Factors

Prime Factors

Accuracy

180 has exactly 18 positive factors

Three distinct prime numbers in factorization

This method guarantees correct count every time

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Perfect Squares and Perfect Cubes

Perfect Squares A number is a perfect square if and only if all exponents in its prime factorization are even.

36 = 2² × 3² 7 (both even) 72 = 2³ × 3²

(3 is odd)

Perfect Cubes A number is a perfect cube if and only if all exponents are multiples of 3.

64 = 2v 7 (6 is multiple of 3) 216 = 2³ × 3³ 7 (both multiples of 3)

GCD and LCM Reinforcement GCD: Product of common prime factors

LCM: Product of all distinct prime

raised to their lowest powers.

factors raised to their highest powers.

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Why Prime Factorization is Crucial for the GMAT Foundation for Number Properties It's the underlying principle for understanding GCD, LCM, number of factors, perfect squares/cubes, and divisibility rules.

Problem Simplification Complex number problems often become manageable when numbers are broken down into their prime components.

Data Sufficiency Prime factorization is frequently tested in DS questions related to properties of integers.

Efficiency It provides a systematic and efficient way to analyze numbers, avoiding guesswork or tedious trial-and-error. www.goalisb.com | Shruti P | ISB, IIM, MBA Expert | contact@goalisb.com | https://www.youtube.com/@Goalisb

Common Pitfalls to Avoid Including 1 as a Prime Factor

Missing a Prime Factor

Mistakes in Exponents

Remember, 1 is NOT a prime number.

Be systematic with your division or factor

Double-check the count of each prime

Never include it in your prime

tree to ensure all prime factors are

factor. A single error changes the entire

factorization.

identified.

factorization.

Not Using Distinct Primes

Forgetting the +1 in Formula

Prime factorization must use only prime

A common error in the number of factors

numbers as bases4never composite

formula: it's (a+1)(b+1)(c+1), not (a)(b)(c).

numbers.

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Interactive Check Your Understanding 1

Prime factorize 252

2

How many positive factors does 252 have?

3

Is 400 a perfect square? Use prime factorization to explain

4

If a number has prime factorization 2_ × 3_ and it has 12 factors, what are possible integer values for a and b?

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Practice Questions & Solutions Question 1: Which number has the most positive factors? a) 36 b) 48 c) 72 d) 100

Answer: c) 72

Analysis

³ (3)(3) = 9 factors 48: 2t × 3¹ ³ (5)(2) = 10 factors 72: 2³ × 3² ³ (4)(3) = 12 factors 100: 2² × 5² ³ (3)(3) = 9 factors 36: 2² × 3²

72 has the most factors with 12 total positive factors.

Question 2: N = 2³ × 3ã × 5² has 48 factors. Find x.

Number of factors = (3+1)(x+1)(2+1) = 48 4 × (x+1) × 3 = 48 12 × (x+1) = 48 x+1 = 4, therefore x = 3

Question 3: Smallest K such that 120K is a perfect square 120 = 2³ × 3¹ × 5¹ (all exponents odd) To make them even, K must supply 2¹ × 3¹ × 5¹ K = 30 (120 × 30 = 3600 = 60²)

Question 5: M has exactly 3 positive factors. Which could be M? For exactly 3 factors, M must be the square of a prime: p² 9 = 3² 7 (factors: 1, 3, 9) 25 = 5² 7 (factors: 1, 5, 25) Answer: e) Both b and d www.goalisb.com | Shruti P | ISB, IIM, MBA Expert | contact@goalisb.com | https://www.youtube.com/@Goalisb

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