Prime factorization breaks a natural number into a product of primes, the pieces that cannot be split any further.
Just as a LEGO model comes 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 the number of divisors appears at once.
Change the number and the factor tree and divisor grid are drawn again for that value.
Intuition — break a number apart
To count every divisor, or later to pick common factors and multiples of two numbers, it is faster to write the number first as a product of primes that cannot be split again. That writing is prime factorization, and repeated primes are grouped as powers. If the primes and exponents match, changing the multiply order still gives one result. Add 1 to each exponent and multiply, and you get the divisor count, so this lesson first finishes splitting all the way down to primes.
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Divisor grid — gold = prime factors, blue = divisors
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On the divisor grid the divisors of the chosen number sit as circles, and the prime ones among them are gold. The factor tree is drawn by splitting downward until every leaf is prime. Repeated primes are grouped as powers in the vertical division, and the GCD and LCM diagram lists the shared primes of two numbers with the smaller and larger exponents. Move the slider and the grid, tree, and list are redrawn for that number; the branches do not grow across the screen.
🧱 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
⑤1 is neither prime nor composite. The smallest prime, 2, is the only even prime
⑥GCD × LCM = a × b is in the next chapter
Prime Factor Tree
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Factor tree — leaves are all primes
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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
It is easy to call 1 a prime, or to think the factorization changes when you divide in another order. 1 has only one divisor, so it is neither prime nor composite, and 2 is the only even prime. Whatever order you divide, the primes and exponents stay the same, so the result is one writing. Do not stop after a single division; keep going until the quotient itself is prime.
Number of Divisors
N = pa × qb → (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
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Find GCD and LCM via prime factorization
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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
Start from the smallest prime and keep dividing until the quotient is prime. Group repeated primes as powers, and when you are asked how many divisors there are, add 1 to each exponent and multiply. Starting from large primes often stalls, so check the small ones first. That order is enough to match the gold-circle list you already saw on the grid.
Example 1
Find the prime factorization of 60.
1
Divide by the smallest prime 2: 60 ÷ 2 = 30, 30 ÷ 2 = 15.
60 = 2 × 2 × 15
2
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.
1
First factor into primes.
72 = 2³ × 3²
2
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
Once a number is written with primes and exponents, the next lesson can take GCD as the smaller shared exponents and LCM as the larger exponents of every prime. The identity GCD×LCM=a×b sits on that same writing. Mix min and max and the two swap, so line the prime lists up and compare exponents side by side.
Prime Factorization Summary
N = p1a_1 × p2a_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
1
From 60 = 2² × 3 × 5, a perfect square needs every prime exponent even.
60 = 2² × 3 × 5
2
Multiply by the odd-exponent primes 3 and 5 to make them even.