Use Factor Pairs, Bounds, and Shared Divisors to Recover Hidden Counts
用因数对、范围与公因数还原隐藏数量
All people, item counts and equal shares in this lesson are positive whole numbers unless a laboratory explicitly allows zero adults or zero leftovers. A hidden number of children, students, winners, or items is often a factor of a known total. Build every factor candidate, apply the story’s bounds and remainder rules, then state whether the answer is unique or whether several cases survive.
Build a factor-pair staircase
Factor pairs come in a predictable order: the first factor rises while the partner falls. Once the factors cross near the square root, every pair has already appeared.
One product, many pairings
If ab = N, then a and b form a factor pair of N. For 420, the staircase begins:
Stop after the smaller factor passes:
Factor-pair laboratory
Read the 420 staircase
Need a hint? Start here
A factor pair records both the number of people and the amount each gets.
How can I check my reasoning?
Stop testing smaller factors after √420. There are 12 unordered pairs. The pair with the smallest difference is 20 and 21.
Use neighboring factor pairs in the 420-apple story
This is Chapter 18, Example 5. The total stays 420 while one factor rises by 1 and the other falls by 2.
original problem
A kindergarten teacher divides 420 apples equally among the children. If one more child joins, each child receives 2 fewer apples. How many children were there originally?
Audit neighboring rows of the factor staircase
Complete the worked solution
Need a hint? Start here
Keep the total fixed when comparing old and new shares.
How can I check my reasoning?
na=(n+1)(a−2) gives a=2n+2, so the share is larger than the people count. The staircase orientation is valid here. 14×30=15×28=420. Original children 14; original share 30; new share 28.
Use a general factor-pair transition laboratory
The same idea works whenever a fixed total is shared by a changed number of people and each share changes by a known amount.
Challenge: T = 330, one person joins, each share falls by 3
Need a hint? Start here
Compare two factor pairs with the same total: people increase by 1 while the share decreases by 3.
How can I check my reasoning?
Try 10×33 and then add one person while subtracting three from the share. 10×33=11×30=330. Original people 10; original share 33; new share 30.
Subtract leftovers, then find a shared divisor
This follows Guided Practice 5. The class size divides every amount actually distributed, and it must be larger than every leftover.
original problem
A teacher has 40 oranges, 200 biscuits, and 120 candies. The teacher gives out as many complete rounds as possible, giving every child one item of that kind per round. Afterward, 4 oranges, 20 biscuits, and 12 candies remain. There are too few of each leftover item to give one more to every child. How many children are in the class?
Oranges
Biscuits
Candies
Shared-divisor step
Every possible class size must be a divisor of 36.
Remainder bound
A remainder must be smaller than the divisor. Because the largest leftover is 20:
Only one common divisor survives.
Complete the worked example practice
Need a hint? Start here
Subtract each leftover first. The class size divides every distributed amount and is greater than every leftover.
How can I check my reasoning?
Distributed amounts are 36,180,108. The class size must divide each and exceed 20, the largest leftover. Their GCD is 36. Its only divisor greater than 20 is 36, so there are 36 children.
Use a general equal-distribution laboratory
A common divisor creates candidates. The largest remainder filters them. Sometimes one answer survives, sometimes several survive, and sometimes the data are impossible.
Analyze totals 50,100,150 with leftovers 2,4,6
These questions use the multiple-candidate preset, even if you change the laboratory. Enter all candidate group sizes separated by commas, in any order.
Need a hint? Start here
Keep every common divisor greater than the largest leftover; more than one survivor means the size is not uniquely determined.
How can I check my reasoning?
The distributed amounts are 48,96,144, with GCD 48. Keep its divisors greater than 6. Five candidates: 8,12,16,24,48. Classification: multiple.
Use repeated count labels to recover the circle size
This follows Chapter 18, Example 6. If the same student says two count labels, the number of students divides the difference between those labels.
original problem
More than 30 but fewer than 50 students stand in a circle and take turns, one student at a time in the same direction. They say consecutive whole-number labels without skipping or restarting. The same student says 30 and 198. How many students are there?
Subtract the labels
The class size must divide 168.
See the full laps
Four complete circuits of 42 students separate the two labels.
Same-speaker laboratory
Complete the worked example
Need a hint? Start here
Subtract the labels of the same speaker. The circle size divides that difference and must satisfy both strict bounds.
How can I check my reasoning?
198−30=168; keep divisors strictly greater than 30 and strictly less than 50. Only 42 works. 168÷42=4 complete circuits.
Assign factor roles using a grouping condition
This follows Guided Practice 6. A product’s factors can represent “number of people” and “trees per person,” but the group structure decides which factor plays which role.
original problem
Students are divided equally into three groups. The teacher plants the same number of trees as each student. Altogether they plant 1,073 trees. How many trees does each person plant?
Factor the total
If there are 37 people including the teacher, then there are:
And:
Role audit
Factor-role laboratory
Every adult and student contributes the same positive whole-number amount. Only students form the equal groups, and each group must contain at least one student. Adults may be zero in this laboratory.
Complete the worked example practice
Need a hint? Start here
The people count includes the teacher. Subtract one before checking whether the students form three equal groups.
How can I check my reasoning?
1073=29×37. Test every people divisor: people minus 1 must be positive and divisible by 3. 37 people, 36 students, 29 trees each, 12 students per group.
Transfer the method and report every valid case
The chapter exercises combine factor roles, square relations, and complete factor enumeration. One problem is unique; another has two valid triples.
original Exercise 10: prize winners
750 yuan is divided equally among n winners. The amount each person receives, measured in jiao, is 12 times the number of winners.
Because n is positive:
original Exercise 13: sum many products at once
Choose one one-digit prime and one one-digit composite. Multiply each of the 4 × 4 = 16 possible pairs once and add the results.
Try recalling the shortcut, then reveal the worked solution
original Exercise 12: enumerate the two-digit triples
Find all two-digit integers a < b < c with even sum and:
State what each complete search proves
Need a hint? Start here
Use jiao in the prize equation, list increasing two-digit factor triples, and use two sums for the prime–composite products.
How can I check my reasoning?
For triples, a,b,c are three separate two-digit numbers multiplied together, with a<b<c. 25 winners. Triples (10,18,22) and (11,15,24) both total 50. The 16 prime–composite products total 17×27=459.
Factor-pair and hidden-count workshop
Correct all eight questions 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 leftovers, subtract first, then use a divisor larger than every remainder. Answers: 12; 14; 30; 36; 5; 42; 29; 2.
Exit ticket
Complete all ten missions, including 5 out of 5 on this exit ticket, to earn the certificate.
Certificate of completion
Hidden-Count Factor Detective
Awarded to Student for building factor pairs, applying bounds, filtering shared divisors, and reporting every valid case.
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?
For the final product sum, use (2+3)(4+6). Answers: 10; 36; 4; 12; 50.