Prime Factorization

Prime factorization breaks a natural number into a product of primes, the pieces that cannot be split any further. Just as you take a LEGO model apart brick by brick, every composite number can be written as a product of primes in exactly one way. Group the repeated primes as powers, add one to each exponent and multiply, and you even get how many divisors the number has. Change the number here and watch the factor tree branch out until every leaf is a prime.

Intuition — break a number apart

Choose number36

🧱 Break It Like LEGO!

①Divisor: a number that divides another evenly (divisors of 12: 1,2,3,4,6,12)

②Prime: only divisors are 1 and itself (2,3,5,7,11,13…)

③Prime factorization: split a natural number into a product of primes

④LEGO bricks = primes; finished build = composite

Prime Factor Tree

Number to factor36

Factorization Procedure

Divide by the smallest prime (2) first

2 → 3 → 5 → 7 → … repeat until quotient = 1

Exponent Notation

72 = 2³ × 3²

Group repeated primes as powers

📝 Vertical Division Layout

①Write the divisor on the left, quotient on the right

②E.g. 72 → 2|72 → 2|36 → 2|18 → 3|9 → 3

③Multiply the left column: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

④The factorization is unique regardless of order

Counting Divisors

Number of Divisors

N = p^a × q^b → (a+1)(b+1)

Multiply (exponent + 1) for each prime

Example: divisors of 72

72 = 2³ × 3² → (3+1)(2+1) = 12

Check: 1,2,3,4,6,8,9,12,18,24,36,72

💡 Why "(exponent + 1)"?

①To form a divisor of 72 = 2³ × 3²

②You can use 2 zero, one, two, or three times → 4 choices

③You can use 3 zero, one, or two times → 3 choices

④4 × 3 = 12 divisors total!

Connecting to GCD & LCM

Number A12

Number B18

GCD

Pick the minimum exponent of each shared prime

Only common primes, with the smallest exponent

LCM

Pick the maximum exponent of every prime

All primes, with the largest exponent

🔍 GCD × LCM = a × b

①GCD × LCM equals the product of the two numbers

②Knowing GCD lets you compute LCM instantly

③Tests frequently use this relationship!

Work It Out

Example 1

Find the prime factorization of 60.

Divide by the smallest prime 2: 60 ÷ 2 = 30, 30 ÷ 2 = 15.

60 = 2 × 2 × 15

Since 15 = 3 × 5, write it with exponents.

60 = 2² × 3 × 5

▸ 60 = 2² × 3 × 5

Divide by the smallest primes in turn, then collect into exponents.

Example 2

Find the number of divisors of 72.

First factor into primes.

72 = 2³ × 3²

Add 1 to each exponent and multiply.

(3 + 1)(2 + 1) = 4 × 3 = 12

▸ 12 divisors

Number of divisors = product of (each exponent + 1).

Exam Key Points

Prime Factorization Summary

N = p₁^a_1 × p₂^a_2 × ...

Every composite has a unique factorization into primes

Grade-7 school exam type

You multiply 60 by a natural number to make a perfect square. What is the smallest such number?

①3

②5

③15

④30

⑤60

▸ ③ 15

From 60 = 2² × 3 × 5, a perfect square needs every prime exponent even.

60 = 2² × 3 × 5

Multiply by the odd-exponent primes 3 and 5 to make them even.

60 × (3 × 5) = 900 = 30², number = 15

🎯 Exam Key Points

①1 is neither prime nor composite

②Smallest prime = 2 (only even prime)

③Factorization is always unique

④#divisors = product of (exponent+1)

⑤GCD × LCM = a × b

GCD and LCM

Was this helpful? Support seegongsik