Reconstruct Consecutive Numbers and Real-World Roles from Products
从乘积还原连续数，并把因数匹配到实际角色
All numbers and ages in these problems are positive whole numbers. Ages are measured in completed years. Prime factorization gives the building blocks. Consecutive-number structure, age ranges, score limits, rank language, and other story clues determine how those blocks must be grouped.
Prime factors are pieces; the story chooses the grouping
A product can have many factor groupings. The mathematical structure and the real-world roles tell us which grouping is meaningful.
Consecutive-number model
Nearby factors must be grouped into numbers that differ by the required step.
Real-world role model
The same three numerical factors can be assigned to roles in different orders. Context must decide the order.
Complete the strategy check
Need a hint? Start here
Use every prime copy once; then check the story conditions.
How can I check my reasoning?
Consecutive whole numbers have gap 1; consecutive odd numbers have gap 2. 1; 2; story conditions and bounds. Factorization alone does not assign ages or ranks.
Worked example 1 — recover four consecutive ages from 5,040
Four students have consecutive ages. Their product is 5,040. Use the prime inventory to build four neighboring whole numbers.
Prime inventory
Test a possible youngest age
Why the estimate helps
Because:
The four ages should be clustered near 8 or 9, not near 2 or 20.
Enter the complete original result
Set the youngest-age slider to your answer as well as filling all three boxes.
Need a hint? Start here
Try four neighboring ages near the fourth root of the product.
How can I check my reasoning?
Set the slider to the youngest age and multiply the four ages back. 7×8×9×10=5040. Youngest 7; oldest 10; product 5040.
original Exercise 5 — three consecutive natural numbers
The product is 120. Search around the cube root, keep the factors consecutive, and verify the exact product.
Factor and regroup
The target is close to:
So test three consecutive values centered near 5.
Candidate comparison
Enter the three numbers in increasing order
General consecutive-product laboratory
The search tests positive whole-number starts from 1 through your chosen maximum, including both ends. Step 1 gives consecutive whole numbers. Step 2 tests both odd and even sequences; check that a reported sequence is odd if the question requires odd numbers.
Need a hint? Start here
Try three neighbors around 5; multiply to verify.
How can I check my reasoning?
4×5×6 uses exactly the prime copies in 120. 4,5,6. Increasing the positive start increases the product, so no other start works.
original Exercise 7 — four students one year apart
The four top jump-rope competitors have ages that, from youngest to oldest, increase by one year each, and their age product is 11,880.
Four ages are consecutive natural numbers. Their product is 11,880.
Move the youngest-age slider
Prime grouping
Complete the age reconstruction
Set the youngest-age slider to your answer as well as filling all three boxes.
Need a hint? Start here
Increasing the youngest age increases every factor.
How can I check my reasoning?
Try 9,10,11,12 and set the slider to 9. Their product is 11880 and their sum is 42. Youngest 9; oldest 12.
Transfer — five consecutive odd numbers
Find five consecutive positive odd numbers with product 328,185. Consecutive odd numbers have step 2, so write them as a, a+2, a+4, a+6, a+8.
Test the first odd number
Use the center
Five consecutive odd numbers are symmetric around the middle number:
The fifth root means the positive number whose fifth power is 328,185. Since 12⁵=248,832 and 13⁵=371,293, the root lies between 12 and 13. This suggests testing a middle odd number near 13; the exact product must still be checked.
Enter the key positions
Set the first-odd-number slider to your answer as well as filling the boxes.
Need a hint? Start here
Odd neighbors differ by 2, not 1.
How can I check my reasoning?
Try 9,11,13,15,17; multiply them and set the first-number slider to 9. Product 328185. First 9; middle 13; largest 17; five terms.
Guided Practice 1 — match 4,365 to age, score, and rank
The worked example says a middle-school student earned a good contest result and that age × score × rank = 4,365.
Age
Score
Rank
Assign the factors to roles
Need a hint? Start here
Apply each stated age, score and rank bound before choosing a grouping.
How can I check my reasoning?
4365=3²×5×97. The prime 97 must be in the score, since age and rank are smaller than 97. Score 97 leaves age×rank=45. The only age divisor from 12–18 is 15, leaving rank 3.
original Exercise 6 — a second age–score–rank product
Xiaoli says that rank × age × exam score = 2,910. Use positive whole-number age 12–18, score 60–100, and rank 1–10, including the endpoints. Rank 1 means first place. These are the same explicit ranges as Mission 6.
Factor and compare
The same age and score as the earlier practice appear, while the rank changes.
Role audit
Enter Xiaoli's information
Need a hint? Start here
Use the same role bounds and multiply all three values back.
How can I check my reasoning?
2910=2×3×5×97. The factor 97 must be the score. Age×rank=30, so age 15 and rank 2 are the only allowed assignment; score 97.
Enumerate role assignments and detect non-uniqueness
A practice problem is not unique merely because one answer looks reasonable. A complete search must reveal every arrangement that satisfies the stated conditions.
Role-assignment laboratory
All role values are positive whole numbers, and both endpoints of each range are included. Keep maximum minus minimum at most 50 for age and 300 for score or rank. Widening score and rank keeps the age bounds 12–18.
Ambiguity detective
Three children have positive whole-number ages from 1 through 9, and the product of their ages is 90. Ages may repeat; rearranging the same three ages does not make a new triple. Determine whether these conditions fix one age sum, or whether different sums are possible.
State what the evidence proves
Need a hint? Start here
Finding one solution is not the same as checking all allowed solutions.
How can I check my reasoning?
Test ages 1 through 9, allowing repeats, and keep product 90. Only (2,5,9) and (3,5,6) work. Their sums are 16 and 14; two triples, multiple possible sums.
Consecutive-product and role-matching workshop
Correct all eight responses 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?
Keep unordered age triples separate from their permutations. Answers: 5; 7; 42; 17; 3; 2; 1; 2.
Objective exit ticket
Complete all ten missions, including 5 out of 5 on this exit ticket, to earn the certificate.
Product Reconstruction & Role-Matching Architect
This certifies that
can reconstruct consecutive numbers from a product, match factors to real-world roles, apply context bounds, and identify when a practice problem is not unique.
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?
5×6×7=210 and 6×7×8×9=3024. Answers: 6; 6; 13; 97; 2.
Teaching notes
The age–score–rank activities explicitly state contextual ranges that the printed original leaves implicit.Test 18 Question 8 is not unique as printed: ages 2, 5, 9 and ages 3, 5, 6 both satisfy the stated product and age bound.