Build Standard Prime Factorizations
建立标准分解质因数式
Split a number until every piece is prime, collect repeated prime factors with exponents, write the primes in increasing order, and multiply back to prove that nothing was lost.
Know what counts as a prime factorization
A factorization may still contain composite pieces. A prime factorization is finished only when every factor is prime. Its standard form collects repeated primes with exponents and lists the prime bases from least to greatest.
Choose the most precise description
Need a hint? Start here
A factor divides without a remainder. A prime has exactly two positive divisors.
How can I check my reasoning?
Then distinguish repeated prime copies from collected powers in increasing order. Factorization but unfinished; expanded prime factorization; standard prime factorization; 1 is neither.
Grow different factor trees for the same number
The worked example introduces the standard form with the number 72. A factor tree may branch in different ways, but every complete tree must end with the same prime leaves.
Multiply them to check 8, then split any composite factor again. Compare with the completed trees below.
Prime-leaf inventory
View both starting splits. The route changes; the prime inventory does not.
Read the completed trees
Need a hint? Start here
Split a composite factor again until every leaf is prime. Count repeated leaves.
How can I check my reasoning?
Both routes end with three copies of 2 and two copies of 3. 72=2^3×3^2; five prime copies; two different prime bases.
Collect repeated primes with exponents
An exponent tells how many copies of a prime are multiplied. It is a count, not another factor to multiply by the base.
Exponent machine
An exponent of zero selects no copies of that prime. If all three exponents are zero, the product is 1; this is an empty product, not a prime factorization of 1.
Live result
Encode 360
Need a hint? Start here
An exponent counts copies, not multiplication by the exponent.
How can I check my reasoning?
For 360, count three 2s, two 3s, and one 5. Exponents 3,2,1; multiply-back result 360.
Use repeated short division
The worked example recommends dividing repeatedly by prime numbers. To keep the standard form easy to read, choose the smallest prime that still divides the current quotient.
Factor 360
Prime keypad
Need a hint? Start here
Divide by one prime at a time. Keep dividing until the remaining quotient is 1.
How can I check my reasoning?
Divide 360 by 2 three times, then by 3 twice, then by 5. Quotients: 180,90,45,15,5,1. Stop at 1: 360=2^3×3^2×5.
Use a general prime-factorization laboratory
Factor any whole number greater than 1, inspect its prime inventory, and verify the result by multiplying the prime powers back together.
Choose a number
Factorization audit
original bridge: 5,040
The chapter immediately uses the factorization of 5,040 in a story about four consecutive ages. This lesson only builds the prime inventory; Lesson 18.3 will regroup those prime pieces to solve the story.
Read the 5,040 audit
Need a hint? Start here
Count distinct bases separately from the total number of prime copies.
How can I check my reasoning?
5040=2^4×3^2×5×7. Distinct bases differ from repeated copies. 4; 2; four distinct primes; 5×7=35.
Factor the four original practice numbers
The chapter exercise asks students to prime-factorize 146, 255, 360, and 400. Enter each result in standard form. Use ^ for exponents and ×, * or x for multiplication, for example 2^3 * 3^2. Write only the factorization, without the original number or an equals sign. List prime bases in increasing order and combine repeated copies. You may omit an exponent of 1.
146two prime bases
255three prime bases
360three prime bases
400two prime bases
Need a hint? Start here
Test divisibility by 2, 3 and 5 first; then multiply your prime powers back.
How can I check my reasoning?
Test small primes repeatedly, then multiply back. 146=2×73; 255=3×5×17; 360=2^3×3^2×5; 400=2^4×5^2.
Catch incomplete and incorrect factorizations
A correct product is not enough if a composite factor remains. A list of primes is not enough if their product no longer equals the original number.
Need a hint? Start here
Check both the product and whether every base is prime.
How can I check my reasoning?
A and D contain composite factors; C gives 36 rather than 72. A incomplete; B correct; C wrong product; D incomplete.
See why the prime inventory is unique
The worked example states that every whole number greater than 1 has one unique standard prime factorization. Different factor trees are different routes to the same prime inventory.
Route 1
Route 2
State exactly what the audit proves
Run the two-route audit before checking these questions.
Need a hint? Start here
Different starting splits should leave the same inventory of prime copies.
How can I check my reasoning?
Each route supplies two 2s, two 3s and one 5. This illustrates uniqueness for 180. 2^2×3^2×5; yes; yes.
Prime-factorization workshop
Use factor trees, exponents, short division, and multiplication-back checks. Correct all eight answers to complete the workshop.
Need a hint? Start here
Choose the strategy first: factor, apply the conditions, and multiply back. Revisit the relevant local hint above.
How can I check my reasoning?
For standard form, combine repeated primes and put bases in increasing order. 2^2×3^2; 5; 3; 3^2×5^2; 4; 120; no; 2×73.
Objective exit ticket
Complete Missions 1–8, earn 8/8 in the workshop, and earn 5/5 on this exit ticket to earn the certificate.
Standard Prime-Factorization Builder
This certifies that
can build prime factorizations with factor trees and short division, write standard exponent form, and verify every result by multiplication.
Need a hint? Start here
Work without the laboratories first. Explain which prime copies or factor pairs your answer uses, then verify the result.
How can I check my reasoning?
252=4×63=2^2×3^2×7. 2^3×3^2; 2; 2^4×5^2; same prime inventory; 2^2×3^2×7.