The Central Idea
Every positive integer greater than \(1\) is either prime or can be written as a product of primes.
For example,
\[ 60=2\cdot2\cdot3\cdot5. \]Using exponents,
\[ \boxed{60=2^2\cdot3\cdot5}. \]This is the prime factorization of \(60\).
What Counts as a Prime Factorization?
A prime factorization must contain only prime factors.
For example,
\[ 72=8\cdot9 \]is a factorization, but it is not a prime factorization because both \(8\) and \(9\) are composite.
Continuing,
\[ 8=2^3 \qquad\text{and}\qquad 9=3^2, \]so
\[ \boxed{72=2^3\cdot3^2}. \]Prime Numbers Are Already Fully Factored
If \(p\) is prime, its prime factorization is simply
\[ \boxed{p}. \]For example, \(17\) is prime, so no nontrivial decomposition into smaller positive factors exists.
Thus its prime factorization is just
\[ 17. \]The Number \(1\) Has No Prime Factorization
The number \(1\) is neither prime nor composite.
It cannot be written as a product of one or more prime numbers, because every prime is greater than \(1\).
Therefore,
\[ \boxed{1\text{ has no prime factorization}}. \]Method 1: Factor Trees
A factor tree begins by splitting a composite number into any convenient factor pair.
Then each composite factor is split again. Continue until every branch ends in a prime number.
For \(84\), one possible start is
\[ 84=7\cdot12. \]Since \(7\) is prime but \(12\) is composite,
\[ 12=3\cdot4 \]and
\[ 4=2\cdot2. \]Therefore,
\[ \boxed{84=2^2\cdot3\cdot7}. \]Different Factor Trees Give the Same Prime Factors
We could begin \(84\) differently:
\[ 84=6\cdot14. \]Then
\[ 6=2\cdot3 \]and
\[ 14=2\cdot7. \]The result is again
\[ \boxed{84=2^2\cdot3\cdot7}. \]The path can change, but the final prime factorization does not.
Method 2: Repeated Division by Primes
Another method repeatedly divides by the smallest convenient prime factor.
Factor \(180\):
\[ 180\div2=90, \] \[ 90\div2=45, \] \[ 45\div3=15, \] \[ 15\div3=5, \] \[ 5\div5=1. \]The divisors used were
\[ 2,\ 2,\ 3,\ 3,\ 5. \]Therefore,
\[ \boxed{180=2^2\cdot3^2\cdot5}. \]Why Repeated Division Works
Each successful division removes one prime factor from the number.
For example,
\[ 180=2(90), \] \[ 90=2(45), \] \[ 45=3(15), \] \[ 15=3(5), \] \[ 5=5(1). \]Multiplying all removed prime factors reconstructs the original number.
Using Divisibility Tests Efficiently
Divisibility tests help identify likely prime factors quickly.
For example, factor \(450\).
Since \(450\) is even,
\[ 450=2(225). \]Since \(225\) ends in \(5\),
\[ 225=5(45). \]Since \(45\) is divisible by \(5\),
\[ 45=5(9). \]Finally,
\[ 9=3^2. \]Therefore,
\[ \boxed{450=2\cdot3^2\cdot5^2}. \]Writing Repeated Prime Factors with Exponents
Repeated prime factors should usually be written in exponent form.
Instead of
\[ 2\cdot2\cdot2\cdot3\cdot3\cdot5, \]write
\[ \boxed{2^3\cdot3^2\cdot5}. \]Exponent notation makes the multiplicative structure easier to read and compare.
Prime Factors Are Usually Written in Increasing Order
The order of multiplication does not change the value:
\[ 2\cdot5\cdot3\cdot2 = 2\cdot2\cdot3\cdot5. \]A standard presentation places primes in increasing order:
\[ \boxed{2^2\cdot3\cdot5}. \]This makes comparisons between factorizations more efficient.
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every integer greater than \(1\) can be written as a product of primes in essentially one way.
The word “essentially” allows only a change in the order of the factors.
Thus
\[ 2\cdot3\cdot5 \]and
\[ 5\cdot2\cdot3 \]represent the same prime factorization.
Why Uniqueness Matters
Unique prime factorization lets us treat prime exponents as structural information about an integer.
For example,
\[ 360=2^3\cdot3^2\cdot5. \]This immediately tells us that \(360\) contains:
- three factors of \(2\);
- two factors of \(3\);
- one factor of \(5\);
- no other prime factors.
Prime Factorization and Divisibility
Prime factorizations make divisibility comparisons precise.
Suppose
\[ a=2^2\cdot3 \]and
\[ b=2^4\cdot3^2\cdot5. \]Every prime factor required by \(a\) appears in \(b\) with at least as large an exponent. Therefore,
\[ \boxed{a\mid b}. \]A Prime-Exponent Divisibility Rule
Suppose positive integers \(a\) and \(b\) have prime factorizations
\[ a=p_1^{\alpha_1}\cdots p_n^{\alpha_n} \]and
\[ b=p_1^{\beta_1}\cdots p_n^{\beta_n}, \]allowing exponent \(0\) for absent primes.
Then
\[ \boxed{ a\mid b \iff \alpha_i\le\beta_i \text{ for every }i }. \]Example: Testing Divisibility from Prime Factors
Determine whether \(72\mid360\).
\[ 72=2^3\cdot3^2, \] \[ 360=2^3\cdot3^2\cdot5. \]The factorization of \(360\) contains every prime power required by \(72\).
Therefore,
\[ \boxed{72\mid360}. \]Example: Detecting Failure of Divisibility
Determine whether \(48\mid360\).
\[ 48=2^4\cdot3, \] \[ 360=2^3\cdot3^2\cdot5. \]The number \(48\) requires four factors of \(2\), but \(360\) contains only three.
Therefore,
\[ \boxed{48\nmid360}. \]Prime Factorization and the GCD
Prime factorization provides a direct way to find a greatest common divisor.
For
\[ 72=2^3\cdot3^2 \]and
\[ 120=2^3\cdot3\cdot5, \]take the smaller exponent of each shared prime:
\[ \boxed{\gcd(72,120)=2^3\cdot3=24}. \]Prime Factorization and the LCM
For the same numbers, take every prime that appears and use the larger exponent:
\[ \operatorname{lcm}(72,120) = 2^3\cdot3^2\cdot5. \]Therefore,
\[ \boxed{\operatorname{lcm}(72,120)=360}. \]Prime Factorization of Powers
If
\[ n=2^a3^b5^c\cdots, \]then raising \(n\) to a positive integer power multiplies every prime exponent by that power.
For example,
\[ 12=2^2\cdot3. \]Therefore,
\[ 12^3=(2^2\cdot3)^3=2^6\cdot3^3. \]Thus
\[ \boxed{12^3=2^6\cdot3^3}. \]Perfect Squares and Prime Exponents
A positive integer is a perfect square exactly when every exponent in its prime factorization is even.
For example,
\[ 900=2^2\cdot3^2\cdot5^2. \]Every exponent is even, so
\[ \boxed{900=30^2}. \]Why Even Exponents Produce Squares
If
\[ n=p_1^{2a_1}p_2^{2a_2}\cdots p_k^{2a_k}, \]then
\[ n= \left( p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k} \right)^2. \]The prime factors can be paired completely, producing an integer square root.
Perfect Cubes and Prime Exponents
A positive integer is a perfect cube exactly when every prime exponent is a multiple of \(3\).
For example,
\[ 1728=2^6\cdot3^3. \]Since \(6\) and \(3\) are multiples of \(3\),
\[ \boxed{1728=12^3}. \]Counting Positive Divisors from Prime Factorization
Prime factorization also reveals how many positive divisors an integer has.
Suppose
\[ n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}. \]A divisor may use any exponent from \(0\) through \(a_i\) for each prime. Therefore the number of positive divisors is
\[ \boxed{ (a_1+1)(a_2+1)\cdots(a_k+1) }. \]Example: Counting Divisors
Consider
\[ 72=2^3\cdot3^2. \]A divisor can contain \(2\) to exponent
\[ 0,\ 1,\ 2,\text{ or }3 \]and \(3\) to exponent
\[ 0,\ 1,\text{ or }2. \]Therefore the number of positive divisors is
\[ (3+1)(2+1)=12. \]Thus
\[ \boxed{72\text{ has }12\text{ positive divisors}}. \]Reconstructing the Original Number
A prime factorization should always multiply back to the original integer.
Suppose we claim
\[ 540=2^2\cdot3^3\cdot5. \]Compute:
\[ 2^2\cdot3^3\cdot5 = 4\cdot27\cdot5 = 540. \]The original number is recovered, so the factorization is correct.
Example: Full Factorization Workflow
Find the prime factorization of
\[ 756. \]Divide by \(2\):
\[ 756=2(378). \]Divide by \(2\) again:
\[ 378=2(189). \]The digit sum of \(189\) is
\[ 1+8+9=18, \]so divide by \(3\):
\[ 189=3(63). \]Continue:
\[ 63=3(21), \] \[ 21=3(7). \]Since \(7\) is prime,
\[ \boxed{756=2^2\cdot3^3\cdot7}. \]Checking the Full Workflow
Reconstruct:
\[ 2^2\cdot3^3\cdot7 = 4\cdot27\cdot7 = 108\cdot7 = 756. \]The check confirms the factorization.
Common Error: Stopping with Composite Factors
Suppose a student writes
\[ 90=9\cdot10. \]This is correct as a factorization, but not as a prime factorization because \(9\) and \(10\) are composite.
Continue:
\[ 9=3^2, \qquad 10=2\cdot5. \]Therefore,
\[ \boxed{90=2\cdot3^2\cdot5}. \]Common Error: Forgetting Repeated Factors
Consider
\[ 48. \]It is not enough to write
\[ 48=2\cdot3. \]The complete prime factorization is
\[ \boxed{48=2^4\cdot3}. \]Each repeated prime factor must be counted.
Common Error: Treating \(1\) as a Prime Factor
The number \(1\) is not prime, so it is not included in prime factorizations.
For example, write
\[ 30=2\cdot3\cdot5, \]not
\[ 30=1\cdot2\cdot3\cdot5. \]Common Error: Adding Prime Factors
Prime factorization uses multiplication.
For example,
\[ 42=2\cdot3\cdot7, \]not
\[ 42=2+3+7. \]The purpose is to express the number as a product of prime building blocks.
Common Error: Incorrect Exponent Compression
Suppose the repeated factors are
\[ 2\cdot2\cdot2\cdot5\cdot5. \]The correct exponent form is
\[ \boxed{2^3\cdot5^2}. \]The exponent counts how many times the same prime appears as a factor.
A Reliable Factor-Tree Procedure
- Start with a positive integer greater than \(1\).
- If it is prime, stop.
- If it is composite, split it into any convenient factor pair.
- Circle or otherwise mark prime factors.
- Continue factoring every composite factor.
- Collect all prime leaves.
- Write repeated prime factors with exponents.
- Multiply the prime powers to verify the original number.
A Reliable Repeated-Division Procedure
- Test the smallest convenient prime divisor.
- Divide when the quotient is an integer.
- Record the prime divisor used.
- Repeat with the quotient.
- Move to the next prime only when the current prime no longer divides.
- Stop when the quotient becomes \(1\).
- Group repeated prime divisors using exponents.
- Multiply to verify.
Practice 1: Factor Tree
Find the prime factorization of
\[ 96. \]Practice 2: Repeated Division
Find the prime factorization of
\[ 150. \]Practice 3: Larger Integer
Find the prime factorization of
\[ 924. \]Practice 4: Prime or Already Finished?
Give the prime factorization of
\[ 131. \]Practice 5: Test Divisibility from Prime Factors
Determine whether
\[ 54\mid756. \]Practice 6: Perfect Square?
Determine whether
\[ 1764 \]is a perfect square by using prime factorization.
Practice 7: Perfect Cube?
Determine whether
\[ 2160 \]is a perfect cube.
Practice 8: Count Positive Divisors
How many positive divisors does
\[ 360 \]have?
Practice 9: Reconstruct the Number
Evaluate
\[ 2^4\cdot3^2\cdot7. \]Practice 10: Identify the Error
A student writes
\[ 108=6\cdot18 \]and says the prime factorization is complete.
Explain what remains to be done.
Practice Answers
Practice 1
\[ 96=2\cdot48 =2^2\cdot24 =2^3\cdot12 =2^4\cdot6 =2^5\cdot3. \]Therefore,
\[ \boxed{96=2^5\cdot3}. \]Practice 2
\[ 150=2(75)=2\cdot3\cdot25. \]Since
\[ 25=5^2, \]we have
\[ \boxed{150=2\cdot3\cdot5^2}. \]Practice 3
\[ 924=2(462)=2^2(231). \]Then
\[ 231=3(77)=3\cdot7\cdot11. \]Therefore,
\[ \boxed{924=2^2\cdot3\cdot7\cdot11}. \]Practice 4
Since
\[ \sqrt{131}\approx11.45, \]test the primes \(2,3,5,7,11\). None divides \(131\).
Therefore \(131\) is prime, so its prime factorization is
\[ \boxed{131}. \]Practice 5
\[ 54=2\cdot3^3 \]and
\[ 756=2^2\cdot3^3\cdot7. \]The factorization of \(756\) contains every prime power required by \(54\). Therefore,
\[ \boxed{54\mid756}. \]Practice 6
\[ 1764=42^2. \]Its prime factorization is
\[ 1764=2^2\cdot3^2\cdot7^2. \]Every exponent is even, so
\[ \boxed{1764\text{ is a perfect square}}. \]Practice 7
\[ 2160=2^4\cdot3^3\cdot5. \]The exponents \(4\) and \(1\) are not multiples of \(3\). Therefore,
\[ \boxed{2160\text{ is not a perfect cube}}. \]Practice 8
\[ 360=2^3\cdot3^2\cdot5. \]Therefore the number of positive divisors is
\[ (3+1)(2+1)(1+1)=24. \]Thus
\[ \boxed{360\text{ has }24\text{ positive divisors}}. \]Practice 9
\[ 2^4\cdot3^2\cdot7 = 16\cdot9\cdot7 = 1008. \]Therefore,
\[ \boxed{1008}. \]Practice 10
Both \(6\) and \(18\) are composite. Continue factoring:
\[ 6=2\cdot3 \]and
\[ 18=2\cdot3^2. \]Therefore,
\[ \boxed{108=2^2\cdot3^3}. \]What to Remember
- A prime factorization expresses an integer greater than \(1\) entirely as a product of primes.
- Prime numbers are already fully factored.
- The number \(1\) has no prime factorization.
- Factor trees and repeated prime division are two standard methods.
- Different valid factorization paths lead to the same final prime factors.
- Repeated prime factors are written efficiently with exponents.
- Prime exponents encode divisibility, GCD, LCM, perfect-power, and divisor-count information.
- Always multiply the prime powers back together to verify the result.
The essential goal is: continue decomposing until every factor is prime, then record the unique prime structure clearly.
Where We Go Next
Prime factorization describes integers through their multiplicative building blocks. The next lesson, Congruence and Modular Arithmetic, shifts attention from factors to remainders and introduces a new way to classify integers according to how they behave under division by a fixed modulus.