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🔬 Prime Factorization

Break any number into its prime building blocks.

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⚖️ GCF & LCM Calculator

Find the Greatest Common Factor and Least Common Multiple of multiple numbers.

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📐 Factor Formulas & Theory

What is a Factor?

A factor of n is any integer that divides n exactly (no remainder).

n ÷ f = whole number → f is a factor of n

Factors of 12: 1, 2, 3, 4, 6, 12
Because: 12÷1=12, 12÷2=6, 12÷3=4, 12÷4=3, 12÷6=2, 12÷12=1

Finding Factors Efficiently

Only test divisors up to √n. Each factor below √n pairs with one above.

For n, test i from 1 to √n:
if n % i === 0:
i is a factor
n/i is also a factor (its pair)

Example: n = 36, √36 = 6
Test: 1→36, 2→18, 3→12, 4→9, 6→6
Result: 1,2,3,4,6,9,12,18,36 (9 factors)

Number of Factors Formula

If n = p₁^a × p₂^b × p₃^c ... then:

Number of factors = (a+1)(b+1)(c+1)...

Example: 360 = 2³ × 3² × 5¹
Factors = (3+1)(2+1)(1+1) = 4×3×2 = 24 factors

Perfect squares always have an odd number of factors.

Sum of Factors Formula

If n = p₁^a × p₂^b then:
Sum = (p₁^(a+1) - 1)/(p₁-1) × (p₂^(b+1) - 1)/(p₂-1)

Example: 12 = 2² × 3¹
Sum = (2³-1)/(2-1) × (3²-1)/(3-1)
= 7/1 × 8/2 = 7 × 4 = 28
Verify: 1+2+3+4+6+12 = 28 ✓

Fundamental Theorem of Arithmetic

Every integer > 1 can be expressed as a UNIQUE
product of prime numbers (up to order).

60 = 2² × 3 × 5
This is the ONLY way to write 60 as prime product.
This is why prime factorization is so powerful!

GCF & LCM Relationship

For any two numbers a and b:
GCF(a,b) × LCM(a,b) = a × b

Example: a=12, b=18
GCF=6, LCM=36
6 × 36 = 216 = 12 × 18 ✓

This means: LCM = (a×b) / GCF

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