Solve Digit-Sum Equations and Build Extremal Numbers
数字和：方程、约束与最值
Turn place value into equations, use digit bounds and modulo 9 to shorten a search, and construct the smallest possible number under digit-sum and parity constraints.
Place value creates the equation. Digit bounds make the search finite.
Turn a numeral into a digit-sum equation
A three-digit numeral is not merely “three symbols.” Its digits carry hundreds, tens, and ones values.
Place value first
For a three-digit number, 1 ≤ a ≤ 9 and 0 ≤ b,c ≤ 9, so 1 ≤ s(N) ≤ 27.
Two high-value invariants
Also, if N = k·s(N), then:
These facts remove impossible digit sums before any full enumeration.
Digit-equation translator
Checkpoint
Need a hint for Mission 1?
Use the fixed numeral195.
Check your method
s(195)=15 and13×15=195. A three-digit digit sum is at most27. Every N−s(N) is divisible by9; N=10 gives exactly9, so no larger universal divisor works. 15;195;27;9.
Smallest means leftmost digits first
First minimize the number of digits: d digits can hold a sum of at most 9d. Once d is fixed, a smaller first digit makes a smaller number regardless of later digits. Choose the smallest first digit that leaves enough capacity in the remaining places, and continue. If the final digit must be even or odd, reserve its capacity before making each choice.
Solve the worked example equation N = 13·s(N)
Use place value, then use digit bounds before testing the remaining cases.
A three-digit number is divisible by 13, and its quotient equals its digit sum.
Complete candidate audit
| c | b=29−4c | Digit b? | Number | 13·s(N) | Result |
|---|
Checkpoint
Need a hint for Mission 2?
29a=b+4c≤45 forces a=1.
Check your method
Then b=29−4c is a digit only for c=5,6,7, giving b=9,5,1. a1;three values;117,156,195;sum468.
Enumerate equations of the form N = k·s(N)
The digit sum is small, so search possible digit sums—not every numeral.
Find the three-digit number satisfying N=18·s(N).
Since N≡s(N) (mod 9):
Thus s(N) is a multiple of 9. For a three-digit number, the only candidates are:
Why this search is complete
A d-digit number has digit sum at most 9d. For each possible sum t, the equation gives only one numerical candidate:
Check its digit length and whether s(N)=t. No other N can work.
General multiplier-equation laboratory
Checkpoint
Need a hint for Mission 3?
After the modulo-9 filter, test sums9,18,27.
Check your method
The candidates162,324,486 have digit sums9,9,18 respectively. Only162 agrees with the sum used to create it. List9,18,27;162;9;one answer.
Use the invariant N − s(N)
This difference is always divisible by 9, but a remainder argument may leave more than one mathematically valid answer.
A four-digit number minus its digit sum is 603□. What can □ be?
Its digit sum is:
Therefore the blank may be:
Wildcard difference laboratory
| Result | Possible final digit | First witness N | s(N) | Check |
|---|
Checkpoint
Need a hint for Mission 4?
Find endings divisible by9, then prove each has a real four-digit witness.
Check your method
Both6030 and6039 are multiples of9. For example6040−10=6030 and6050−11=6039. List0,9;any valid witness for6030;any valid witness for6039;two endings.
Build the smallest number with a fixed digit sum
Minimize the number of digits first. Then make the leading digit as small as possible.
Find the smallest positive integer whose digit sum is 2003.
General smallest-number constructor
Checkpoint
Need a hint for Mission 5?
Use the fixed target digit sum2003.
Check your method
222 digits hold at most1998, so223 are necessary. The first digit must be at least5 to leave enough capacity in222 remaining places. 223 digits;leading5;222 trailing9s;1998.
Build the smallest odd or even number
A parity condition controls the final digit, so the unrestricted minimum may no longer work.
Smallest odd number with digit sum 44
Four digits can total at most 36, so five digits are required. The leading digit must be at least 8:
Smallest odd and even numbers with digit sum 22
4+9+9=22.
A leading 4 would force 499, which is odd. The next possible leading digit is 5.
Parity-constrained constructor
Checkpoint
Need a hint for Mission 6?
The final digit determines parity; minimize from left to right.
Check your method
For sum22,499 is smallest and odd. An even last digit is at most8, so a three-digit even answer must start at least5:598. 89999;499;598;five digits.
Use several original constraints at once
One simple equation may identify three consecutive numbers; the remaining equations then become a complete verification audit.
“Welcome,” “Spring,” and “Cup” represent three consecutive positive whole numbers.
One printed equation is:
Write the three consecutive values in descending order as:
The sum fixes the set {1,2,3}, but does not assign symbols. From Welcome−Spring−Cup=0 and their sum 6, Welcome=3. From Welcome+Spring−Cup=4, Spring−Cup=1, so Spring=2 and Cup=1. Now verify all eight equations.
Check all eight printed equations
Checkpoint
Need a hint for Mission 7?
First find the set of values, then assign each name.
Check your method
The sum6 forces consecutive values1,2,3. The difference equation forces Welcome3, then Spring2 and Cup1. 3;2;1;eight equations.
Enumerate mixed conditions and preserve every answer
When several conditions are combined, a complete list is stronger than one lucky example.
Rank 2012 in the increasing list of positive multiples of 4 with digit sum 5, counting the first as position 1
Lucky pairs
The two integers differ by 2, and both digit sums are divisible by 7. Both numbers must be in 100–200, including the endpoints. The complete pairs are:
Rank laboratory
Lucky-pair laboratory
Checkpoint
Need a hint for Mission 8?
Use the fixed2012 rank and100–200 lucky-pair conditions.
Check your method
Count qualifying positive multiples of4 in increasing order through2012. The valid pairs are149,151 and158,160; either order within a pair is accepted. Rank12;two pairs;enter either complete pair.
Try a new problem before the workshop
Find the smallest positive even number with digit sum 14. Explain why it is smallest.
Compare your reasoning after trying
Two digits suffice. The units digit is at most 8, so the tens digit is at least 6. The answer is 68.
Independent workshop
Correct all eight answers to complete the workshop.
Optional reflection — not automatically graded
Need a hint for Mission 9?
Lists and named assignments have different ordering rules.
Check your method
The blank permits0 and9; named values must be Welcome3,Spring2,Cup1 in that order. 3;468;162;list0,9;223;598;3,2,1;12.
Objective exit ticket
Answer all five correctly and complete the other nine missions to unlock the certificate.
Digit-Sum Equation & Extremal-Number Architect
This certifies that Student completed all ten missions and can translate place value into equations, use modulo 9 responsibly, build extremal numbers, and preserve every valid solution.
Chapter 21 • Digit Sums
Need a hint for Mission 10?
Use the worked example simplified equation29a=b+4c.
Check your method
The greatest universal divisor is9. For digit sum16, an even ones digit is at most8, so the tens digit is at least8. b+4c (spaces and × or * allowed);9;499;222;88.