Decide Whether a Number Is Prime or Composite
质数与合数：判断一个数属于哪一类
Build factor pairs, learn why a primality test stops at the square root, create the primes below 100 with a sieve, and investigate the worked example’s six large candidate numbers.
Prime, composite, or neither?
This lesson classifies nonnegative whole numbers: 0, 1, 2, 3, and so on. The words describe the number of positive divisors—not whether a number “looks complicated.”
Prime
A whole number greater than 1 with exactly two positive divisors:
Examples: 2, 3, 5, 7, 11.
Composite
A whole number greater than 1 with more than two positive divisors.
Example: 15 = 3 × 5.
Neither
The numbers 0 and 1 are neither prime nor composite.
The number 1 has only one positive divisor, not two.
Live divisor classifier
Check the definitions
Need a hint? Start here
A positive divisor divides with no remainder. Test 1 and the number itself first.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Use factor pairs as evidence
A factor pair multiplies to the number. A pair other than 1 × n is a direct witness that the number is composite.
Factor-pair explorer
What counts as proof?
For example:
For a prime, however, finding no factor by casual guessing is not enough. We need a complete stopping rule. Mission 3 builds that rule.
Read the factor evidence
Need a hint? Start here
Find a factor pair other than 1 × n. For 91, try the primes 2, 3, 5, then 7.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Know when a primality test may stop
Factor pairs come in a small factor and a large factor. Once the small side has passed the square root, every possible pair has already been checked from the other direction.
Prime-divisor trial list
Why the stop is safe
The square root √n is the nonnegative number whose square is n. For example, √121=11 because 11×11=121. Always include the boundary itself: testing only primes below 11 would miss the factor of 121.
If n = a × b and both a and b were greater than √n, then their product would be greater than n. That is impossible.
For 97:
Only the prime divisors 2, 3, 5, and 7 need to be tested.
Use the stopping rule
Need a hint? Start here
Check prime factors up to and including the square root. A larger factor must have a smaller partner already covered.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Build every prime below 100 with a sieve
The worked example asks students to remember the 25 primes below 100. A sieve lets you construct the list instead of treating it as unexplained memorization.
Sieve of Eratosthenes
Begin by crossing multiples of 2 greater than 2.
The 25 primes below 100
Only 2 is even. Every other prime in the list is odd.
Audit the completed sieve
Apply all four crossing steps, or reveal the final sieve, before checking these answers.
Need a hint? Start here
Cross out multiples of each base prime, keeping the prime itself. Every composite up to 100 has a prime factor at most 10.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Use a complete primality checker
Quick tests reject many composite numbers. When no quick test works, the checker tests every prime divisor up to the square root and shows the evidence.
Number to investigate
Evidence ledger
Classify with evidence
Need a hint? Start here
A failed small-prime test supplies a factor; a passed test alone does not prove primality.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Investigate the worked example’s six large candidates
Guided Practice 2 lists six numbers and states that exactly one is prime. Use quick filters first, then a complete square-root audit for the survivor.
original-based investigation
Classify:
Record the worked example result
Run the full six-number audit, then answer all three questions to complete this mission.
Need a hint? Start here
Use final digits and digit sums first. The remaining candidate needs all prime divisors up to its square-root boundary checked.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Search the worked example’s prime reference
This searchable table lists the first 1,000 primes, ending at 7,919. Look up a prime by its position in the list or investigate a value.
Look up a prime by index
Investigate a value
Reference facts
The table is a reference, not a substitute for proof. When a problem asks why a number is prime, use the square-root test.
Use the reference accurately
Need a hint? Start here
A prime table is a reference. Knowing how to justify one entry matters more than memorizing distant entries.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Transfer the definition to original exercises
Use digit sums, parity, and the definition itself before launching a large search.
Can a permutation of 1–9 be prime?
Find a prime triple
To prove you found every answer, consider division by 3. If p leaves remainder 1, then p+14 is a multiple of 3 greater than 3. If p leaves remainder 2, then p+10 is such a multiple. Thus the only possible prime p is 3; verify all three numbers. A search only up to 200 would not by itself prove this global claim.
Find a prime p such that all three numbers are prime:
Reminder: 0 and 1
The chapter introduction defines 0 and 1 as neither prime nor composite.
Transfer check
Need a hint? Start here
Choose the rule before calculating: parity, digit sum, or trial division.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Prime-or-composite workshop
Correct all eight questions to complete the workshop.
1. The number 1 is:
2. What is the only even prime?
3. What is the smallest composite natural number?
4. How many primes are below 100?
5. What is the largest prime below 100?
6. Which candidate is prime: 31415, 314159, or 31415927?
7. Every arrangement of digits 1 through 9 is divisible by which one-digit composite number?
8. Find p if p, p+10, and p+14 are all prime.
Need a hint? Start here
Use the definition for 0 and 1, and supply a factor to disprove primality.
How to check my reasoning
Write down the conditions first. Show one calculation or rejected case, and check that your result satisfies every condition.
Exit ticket
Complete Missions 1–8, earn 8/8 in the workshop, and earn 5/5 here to unlock your certificate. The optional reflection is not automatically graded.
1. A prime has exactly how many positive divisors?
2. How should 0 and 1 be classified?
3. Is 97 prime or composite?
4. Is 121 prime or composite?
5. What is the largest prime you need to test as a possible divisor of 113?
Optional reflection — not automatically graded
Certificate of completion
Prime & Composite Number Detective
has completed Lesson 17.1 and demonstrated accurate use of definitions, factor pairs, square-root primality tests, the sieve, and prime references.
Teaching notes
The original candidate list and prime-reference endpoints are preserved.The square-root proof, generated sieve, explicit factor witnesses, searchable table, feedback, workshop, and exit ticket are added instructional scaffolds.
Need a hint? Start here
For the new question, compare 7² and 11² with 113. Which primes remain below the boundary?
How to check my reasoning
Explain your method in the reflection box. The checker grades the answer; your explanation is for you to review.