Find a Hidden Divisor from the Remainder
由余数反求隐藏的除数
Subtract the remainder, factor what remains, and filter the factors by every condition in the story. Then verify each surviving divisor in the original division.
Learn it on your own
Work through one or two missions at a time. First predict an answer on paper, then use a laboratory to test it. If stuck, open the local hint and then the reasoning. Try the workshop yourself before choosing a worked review after an attempted check. Every explanation you need is on this page; external chat is optional.
Always check: whole-number quantities, positive divisor, and 0≤r<d. A zero quotient is possible when N<d; a zero remainder means exact division. If a question asks for positive numbers, exclude zero.
Peel off the remainder
A remainder statement hides a complete multiple of the divisor. Remove the leftover first, and the divisor must divide everything that remains.
Start from the division story
A divisor is the number we divide by; a remainder may be left. Saying that a number divides another number means it is a factor, with no remainder. Here d need not be a factor of N, but it is a factor of N − r.
Subtract the remainder from both sides:
Remainder peeler
Read the worked example doorway
Need a hint? Try this first
Remove the leftovers.
Read the reasoning, then explain it yourself
310−37=273 must consist of complete groups: d×q. Also d>37.
Build a factor-and-filter sieve
Factors create the candidate pool. The remainder bound, digit length, and any extra story condition remove candidates that cannot be divisors.
Subtract
Calculate N − r.
Factor
List every positive factor of N − r.
Filter
Keep factors with d > r and the required size.
Verify
Check N = d × q + r in the original problem.
Factor sieve for the worked example
Choose the rules that belong in the sieve
Need a hint? Try this first
Apply every restriction.
Read the reasoning, then explain it yourself
A two-digit divisor is 10 through 99, divides N−r, and is greater than r. A factor alone is not enough.
Solve the worked example completely
310 is divided by an unknown two-digit positive whole number, leaving remainder 37. The worked example asks for every possible two-digit divisor.
Step A · Make a multiple
So the divisor must be a factor of 273.
Step B · List the factors
21 is a factor
It fails the remainder rule, so it cannot be the divisor in this division.
39 survives
The remainder 37 is smaller than 39.
91 survives
The remainder 37 is smaller than 91.
Rebuild the result yourself
Select every two-digit candidate that gives remainder 37 when dividing 310. Select no other candidate.
Need a hint? Try this first
List factor pairs of 273.
Read the reasoning, then explain it yourself
1×273, 3×91, 7×39, 13×21 exhaust the pairs. Two-digit factors above 37 are 39 and 91; their quotients are 7 and 3.
Verify survivors and reject almost-candidates
A factor can still fail. Always compare the claimed remainder with the divisor and then check the actual division.
Candidate verifier: check the remainder
This tool checks the remainder condition. Check size restrictions separately: 273 gives the requested remainder but is not a two-digit answer to the practice problem.
Why the tempting equation fails
This equality is true, but it is not a finished division statement because:
The actual division
The valid remainder is 16 because:
Audit the candidates
Need a hint? Try this first
Test an actual division.
Read the reasoning, then explain it yourself
310=21×14+16, so 21 does not give remainder 37. But 310=39×7+37=91×3+37.
Guided Practice 1: find every divisor
A two-digit divisor is used to divide 1,477, leaving remainder 49. The worked example asks for every possible two-digit divisor.
Subtract the remainder
The divisor must be a two-digit factor of 1428 that is greater than 49.
Surviving factors
| Divisor d | Quotient q | Verification |
|---|---|---|
| 51 | 28 | 1477 = 51 × 28 + 49 |
| 68 | 21 | 1477 = 68 × 21 + 49 |
| 84 | 17 | 1477 = 84 × 17 + 49 |
Record the complete candidate list
Need a hint? Try this first
First calculate 1477−49.
Read the reasoning, then explain it yourself
1428=2×2×3×7×17. The two-digit factors above 49 are 51, 68 and 84. Rebuild 1477 with each candidate.
Use the hidden-divisor laboratory
Change the dividend, remainder, and allowed divisor range. The laboratory lists every factor, applies every filter, and classifies the result.
This search uses a positive quotient, so enter N>r. If q=0, then N=r, and any divisor greater than r works; that is a separate family rather than a factor list of a positive number.
General factor sieve
Read the 474-remainder-6 result with divisors from 10 through 99
Need a hint? Try this first
Keep all two-digit factors greater than 6.
Read the reasoning, then explain it yourself
474−6=468. The complete list is 12,13,18,26,36,39,52,78. There are eight; the greatest is 78.
Let the story add a final filter
Arithmetic may leave several divisors. A story condition—such as “fewer than 60 students”—can select the one that makes sense.
16 · Question 1
A class of fewer than 60 students buys 310 workbooks. Each student receives as many whole workbooks as possible, with equal numbers for everyone and 37 workbooks left over. How many students are in the class?
Exercise 1
Find every two-digit divisor that gives remainder 4 when dividing 109.
Apply both story filters
Select every two-digit number that gives remainder 4 when dividing 109.
Need a hint? Try this first
Use the extra story limit last.
Read the reasoning, then explain it yourself
For 310 with remainder 37, the candidates are 39 and 91; fewer than 60 selects 39. For 109 with remainder 4, factor 105: the two-digit candidates are 15,21,35.
Classify unique, multiple, and impossible cases
In each case below, the unknown divisor must be a two-digit positive whole number. The same method may leave one candidate, several candidates, or none. Name exactly what the evidence proves.
5122, remainder 66
The two-digit factors are 16, 32, 64, and 79. Only 79 is greater than 66.
474, remainder 6
Eight two-digit factors survive:
100, remainder 60
Every two-digit factor of 40 is at most 40, but a valid divisor would have to be greater than 60.
Classify the three cases
Need a hint? Try this first
Count the surviving candidates.
Read the reasoning, then explain it yourself
5122−66=5056=64×79; only 79 is two-digit and greater than 66. The 474 problem has eight answers. For 100 with remainder 60, a divisor must divide 40 yet exceed 60: impossible.
Hidden-divisor workshop
Use Subtract → Factor → Filter → Verify. Correct all eight responses to complete the workshop.
Workshop strategy reminder
Write the division equation or identify the repeated change first. Use the nearby mission hints for the matching method. For “all possible” answers, list the candidates systematically and explain why no other candidate can pass. Check every answer in the original story.
Exit ticket
Try these questions with the examples covered. Complete Missions 1–8, earn 8/8 in the workshop, and earn 5/5 here to unlock your certificate. Reflection is optional and ungraded.
Optional reflection — not automatically graded
Certificate of completion
Hidden-Divisor Factor Detective
has completed Lesson 16.2 and can subtract a remainder, factor the adjusted dividend, apply all divisor restrictions, and verify every survivor.
Worked explanation for the exit ticket
216=2³×3³. Its two-digit factors are 12,18,24,27,36,54,72: seven in all. Each is greater than 5. Only 12 is below 15. For example 221=12×18+5.
If you used this explanation, close it and solve the questions again on paper. Explain why each remainder is smaller than its divisor.
Further practice
The central method and the problems involving 310 with remainder 37 and 1,477 with remainder 49 follow Chapter 16, Example 1 and Guided Practice 1 on this lesson page 97. The exercises involving 109 with remainder 4 and 5,122 with remainder 66 come from the chapter practice on this lesson page 100. The class-size problem and the complete two-digit-divisor problem for 474 with remainder 6 come from Test 16 in this exercise.