Test Squares with Last Digits and Digit Patterns
用末位数字和数字结构检验完全平方数
Reject impossible squares quickly, inspect the last two digits, translate repeated-digit numerals into equations, and enumerate every surviving pattern instead of trusting a guess.
Build the complete units-digit rule
The final digit of a square depends only on the final digit of its square root. Test all ten possible root endings; do not memorize an incomplete list.
Before exploring the wheel, predict the units digit of the square of a number ending in 3.
Your prediction is checked with this mission. Try it before reading the worked steps below.
Choose a root’s final digit
The finished rule
A perfect square can end only in 0, 1, 4, 5, 6, or 9. It can never end in 2, 3, 7, or 8.
Rule checkpoint
Need a hint? Start here
Square each of the ten possible root endings and collect different final digits.
The different possible endings are 0,1,4,5,6,9. The impossible endings add to 2+3+7+8=20. Prediction 9. Checkpoint: 20; six; impossible; 4.
Worked example 3: reject a huge-looking expression without evaluating it
The symbol n! means 1 × 2 × … × n; for example, 4! = 24. The worked example expression is 1! + 2! + 3! + 4! + 5! + 6!. Only its units digit is needed.
Track only the units digit of each term
Every factorial from 5! onward contains both a factor 2 and a factor 5, so it ends in zero.
Factorial-sum units-digit laboratory
Checkpoint for the fixed practice problem above
Need a hint? Start here
Keep only each term’s units digit. A factor 2 and a factor 5 together make a final zero.
The endings 1,2,6,4,0,0 add to 13, so the sum ends in 3. 4; 0; 3; not a perfect square.
Know what a last-digit test can and cannot prove
An impossible final digit proves “not square.” An allowed final digit only means “keep investigating.”
A decisive rejection
Because 7 is impossible as a square’s units digit, 247 cannot be a perfect square. No further work is needed.
A necessary test, not a proof
The units digit 1 is allowed, but:
So 21 is not a square.
Square evidence scanner
Evidence checkpoint
Need a hint? Start here
An impossible final digit rejects a square. An allowed one still needs an exact-root check.
247 is ruled out by 7. For 546, compare 23²=529 with 24²=576. No; 247; 529; 576.
Inspect the last two digits
A square’s final two digits repeat every 100 possible root endings. This stronger test can resolve questions that the units digit cannot.
All square endings (remainders after division by 100)
“Modulo 100” means the remainder after division by 100. Write it as two digits, including a leading zero when needed: 4 has ending 04.
Square the root endings 00 through 99 and keep each distinct result.
There are 22 possible final-two-digit endings.
original Exercise 11
If the tens digit of n² is 7, then its final two digits lie from 70 through 79.
Only one ending survives:
Last-two-digit checker
Three-digit squares with tens digit 7
The complete search through 10² to 31² gives:
Two-digit checkpoint
Need a hint? Start here
Try root endings 00 through 99. Keep the square endings that match the required tens digit.
Only ending 76 survives. Roots 24 and 26 give 576 and 676. 22 different endings; units digit 6; 576; 676.
Worked example 4: find the repeated-pair square
Find every four-digit square whose first two digits match each other and whose last two digits match each other. The notation aabb names the four-digit numeral, not a product: a is a digit 1–9, and b is a digit 0–9. The letters may represent the same digit.
Write the numeral algebraically
If a square contains a factor 11, its prime factorization must contain 11². Therefore:
For 100a+b, the alternating digit sum is a−0+b. It must be a multiple of 11. Since 1≤a≤9 and 0≤b≤9, we have 1≤a+b≤18; the only multiple of 11 in that range is 11. Therefore:
Audit the eight candidates
| 100a+b | ÷ 11 | Quotient square? | Repeated-pair number |
|---|
The surviving structure
Repeated-pair square search
The runtime search checks all 90 values aabb with a = 1…9 and b = 0…9.
Checkpoint for the fixed practice problem above
Need a hint? Start here
Expand the numeral using place value. A prime that divides a square must appear an even number of times.
a+b=11 gives eight candidates. Only (100a+b)/11=64 is square, at a=7,b=4. 7; 4; 7744; root 88.
Guided Practice 3: build a square from square digits
The tens and ones digits must themselves be one-digit perfect squares.
Square digits
The one-digit perfect-square values are:
A two-digit numeral cannot begin with 0.
Check the two-digit squares
Only one square has both digits in {0,1,4,9}:
Three-digit transfer
Find every three-digit square whose hundreds digit plus ones digit equals its tens digit.
Digit-filter checkpoint
Need a hint? Start here
List the two-digit squares, then test each digit against the square-digit set.
Only 49 among 16,25,36,49,64,81 uses two square digits. The three-digit transfer gives 121 and 484. 49; root 7; two transfer solutions; larger 484.
Translate digit patterns into equations
Place value turns a digit puzzle into a short algebraic search. It also reveals when the worked example wording allows more than one answer.
A two-digit number equals ones-digit squared plus tens digit
Let the number be ab, with tens digit a from 1–9 and ones digit b from 0–9. The overline marks a numeral, not multiplication.
The unique two-digit solution is:
A two-digit number plus its reversal is a square
Here ab means 10a+b and ba means 10b+a. Count each original number separately: 29 and 92 are two answers, even though they form one reversed pair.
The sum is at most 198. A square divisible by 11 must contain 11², so the only possible square is:
Hence:
Digit-equation checkpoint
Need a hint? Start here
Write ab as 10a+b and ba as 10b+a. These are numerals, not products.
9a=b(b−1) gives a=8,b=9. For reversal, a+b=11 gives 29,38,47,56,65,74,83,92. 89; digit sum 11; eight numbers; four with a<b.
Use a complete pattern-search studio
Choose a pattern. The page enumerates the full candidate range, applies the exact condition, and reports every match.
Why enumeration matters
A pattern may have:
- one result, such as the repeated-pair square;
- several results, such as the reversed-digit family;
- no result.
Only a complete search or a proof can justify “unique.”
Practice recap
Search-studio checkpoint
Need a hint? Start here
Check the whole allowed root or digit range before claiming a result is unique.
There are 22 square endings; 7744 is the only aabb square; 121 and 484 satisfy the three-digit sum rule. 22; 1; 2; smallest three-digit tens-7 square 576.
Last-digit and digit-pattern workshop
Correct all eight answers to complete the workshop.
Optional reflection — not automatically graded
Need a hint? Start here
Choose a method before calculating. Use the earlier local hint for that method, then verify the root, all stated conditions and the endpoints.
A square with tens digit 7 has units digit 6. A reversal solution must have digit sum 11. Yes; 20; 3; 6; 7744; 49; 89; 4.
Exit ticket
Complete all ten missions, including all five exit questions, to earn your certificate.
Fresh challenge: use the methods from this lesson on these new values. Check your calculations by squaring, listing a short cycle, or verifying every condition.
Square Digit-Pattern Detective
This certifies that a determined mathematician can use units digits, last-two-digit residues, place-value equations, and complete searches to test perfect-square patterns.
Lesson 20.3 • Chapter 20
Need a hint? Start here
Choose a method before calculating. Use the earlier local hint for that method, then verify the root, all stated conditions and the endpoints.
The four reversal numbers with a>b are 65,74,83,92. No; 6; 3; 121; four.
original and instructional-scope note
Dynamic enumeration and the general scanners are added instructional scaffolding.