This MCQ module is based on: Number Line, Digits and Palindromes
TOPIC 9 OF 41
Number Line, Digits and Palindromes
🎓 Class 6
Mathematics
CBSE
Theory
Ch 3 — Number Play
⏱ ~35 min
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This mathematics assessment will be based on: Number Line, Digits and Palindromes
Targeting Class 6 level in Number Theory, with Basic difficulty.
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3.3 Patterns of Numbers on the Number Line
Placing numbers on a number line? helps us see how big they are, what's nearby, and how they compare.
The numbers 2180, 2754, 1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300, 8400 placed on a 1000-step number line.
Figure it Out (Section 3.3)
Identify the points marked on the number lines below, and label the remaining positions.
(a) between 2010 and 2020, one tick is at 2015; (b) between 9996 and 9997; (c) 15,077, 15,078, 15,083; (d) 86,705 and 87,705.
(a) between 2010 and 2020, one tick is at 2015; (b) between 9996 and 9997; (c) 15,077, 15,078, 15,083; (d) 86,705 and 87,705.
(a) Tick points between 2010-2020 are 2011, 2012, …, 2019.
(b) Between 9996 and 9997, halfway = 9996.5.
(c) Between 15,077 and 15,083 the missing integers are 15,079, 15,080, 15,081, 15,082.
(d) Between 86,705 and 87,705 each unit = 1. Missing = 86,706 … 87,704 (in steps of 100 labels = 86,805; 86,905; 87,005; …).
Smallest: circle 2011 / 9996 / 15,077 / 86,705. Largest: box 2019 / 9997 / 15,083 / 87,705 as per each row.
(b) Between 9996 and 9997, halfway = 9996.5.
(c) Between 15,077 and 15,083 the missing integers are 15,079, 15,080, 15,081, 15,082.
(d) Between 86,705 and 87,705 each unit = 1. Missing = 86,706 … 87,704 (in steps of 100 labels = 86,805; 86,905; 87,005; …).
Smallest: circle 2011 / 9996 / 15,077 / 86,705. Largest: box 2019 / 9997 / 15,083 / 87,705 as per each row.
3.4 Playing with Digits
Let's count how many numbers have 1 digit, 2 digits, 3 digits, and so on.
| 1-digit (1–9) | 2-digit (10–99) | 3-digit (100–999) | 4-digit (1000–9999) | 5-digit (10000–99999) |
|---|---|---|---|---|
| 9 | 90 | 900 | 9000 | 90000 |
Digit sums of numbers
Komal observes: when she adds the digits of a number, the digit sum? tells her something. For 68, digit sum = 6 + 8 = 14. For 176, digit sum = 1 + 7 + 6 = 14. For 545, digit sum = 5 + 4 + 5 = 14.
Figure it Out
Q1. Digit sum 14.
(a) Write other numbers whose digits add up to 14.
(b) What is the smallest number whose digit sum is 14?
(c) What is the largest 5-digit number whose digit sum is 14?
(d) How big a number can you form having digit sum 14? Can you make an even bigger number?
(a) 248, 356, 653, 815, 833, 923, 1445, 12335, 23351 … (many possible).
(b) 59 (5 + 9 = 14).
(c) 95000 (9 + 5 + 0 + 0 + 0 = 14).
(d) As large as we wish. E.g. 95, 9005, 900005, 9000000005, … Any number with two non-zero digits (9 and 5) separated by as many zeroes as we like.
(b) 59 (5 + 9 = 14).
(c) 95000 (9 + 5 + 0 + 0 + 0 = 14).
(d) As large as we wish. E.g. 95, 9005, 900005, 9000000005, … Any number with two non-zero digits (9 and 5) separated by as many zeroes as we like.
Q2. Find out the digit sums of all the numbers from 40 to 70.
40→4, 41→5, 42→6, 43→7, 44→8, 45→9, 46→10, 47→11, 48→12, 49→13, 50→5, 51→6, 52→7, 53→8, 54→9, 55→10, 56→11, 57→12, 58→13, 59→14, 60→6, 61→7, 62→8, 63→9, 64→10, 65→11, 66→12, 67→13, 68→14, 69→15, 70→7.
Observation: Within each decade, digit sum increases by 1; at a decade change (e.g. 49→50), it drops by 8.
Observation: Within each decade, digit sum increases by 1; at a decade change (e.g. 49→50), it drops by 8.
Q3. Calculate the digit sums of 3-digit numbers whose digits are consecutive (e.g. 345). Do you see a pattern? Will it continue?
123→6, 234→9, 345→12, 456→15, 567→18, 678→21, 789→24. Pattern: Each sum is a multiple of 3 and increases by 3 each time. Continue? There is no next 3-digit consecutive case after 789, so the sequence stops for 3-digit numbers.
Digit Detectives
Digit '7' occurrences from 1 to 100: 7, 17, 27, 37, 47, 57, 67, 77 (counts as 2), 87, 97 and 70, 71, …, 79 (counts 10, including 77). Total = 20 times. From 1 to 1000: by symmetry, each digit (1–9) appears equally often → 1 + 2 + 3 = 6? No: the digit 7 appears 300 times from 1 to 1000.
3.5 Pretty Palindromic Patterns
Definition
A palindrome? is a number (or word) that reads the same from left-to-right and right-to-left. Examples: 66, 848, 575, 797, 1111. "MALAYALAM" is a word palindrome.
All 3-digit palindromes have the form aba where a = 1..9, b = 0..9 → total 9 × 10 = 90 palindromes. Some: 121, 313, 222, 787, 999.
Reverse-and-add palindromes
Start with a 2-digit number. Add it to the number formed by reversing its digits. If the result is a palindrome, stop. Otherwise, reverse the new number and add again. Repeat.
| Start | Steps | Palindrome |
|---|---|---|
| 34 | 34 + 43 = 77 | 77 |
| 29 | 29 + 92 = 121 | 121 |
| 48 | 48 + 84 = 132; 132 + 231 = 363 | 363 |
| 76 | 76 + 67 = 143; 143 + 341 = 484 | 484 |
Puzzle (p. 62): "I am a 5-digit palindrome. I am an odd number. My 't' digit is double of my 'u' digit. My 'h' digit is double of my 't' digit. Who am I?"
Let u = t÷2 and h = 2t. Also palindrome condition means digits are (t-h pattern). Working out: try t = 2, u = 1, h = 4 → digits t.h.m.h.t with u=1 means … after verification, answer 36,963 or similar. A standard solution is 48,284? Let's re-verify: palindrome form is a b c b a where (a, b, c) ↔ (u, t, h). If u = 1, t = 2, h = 4, we get 12421 — which is a palindrome, odd (ends in 1), u=1, t=2=2×1 ✓, h=4=2×2 ✓. Answer: 12,421.
Let u = t÷2 and h = 2t. Also palindrome condition means digits are (t-h pattern). Working out: try t = 2, u = 1, h = 4 → digits t.h.m.h.t with u=1 means … after verification, answer 36,963 or similar. A standard solution is 48,284? Let's re-verify: palindrome form is a b c b a where (a, b, c) ↔ (u, t, h). If u = 1, t = 2, h = 4, we get 12421 — which is a palindrome, odd (ends in 1), u=1, t=2=2×1 ✓, h=4=2×2 ✓. Answer: 12,421.
Activity: Predict → Observe → Explain — Reverse and Add
Predict: For every 2-digit number, does the reverse-and-add process always eventually give a palindrome?
- Pick any 2-digit number (say 29).
- Add it to its reversal (29 + 92 = 121).
- If it is a palindrome, stop. Otherwise repeat.
- Try 10 different starting numbers and record the number of steps.
Observe: All 2-digit numbers reach a palindrome, most in 1–6 steps. Explain: Because digit-reversal preserves the digit sum, but symmetrises the positional value — repeating this smoothens digit differences. The famously stubborn example is 196 (3-digit), which has been checked to millions of iterations without producing a palindrome — known as a Lychrel candidate.
Competency-Based Questions
Scenario: Asha is preparing prize cards with numbers that have digit sum 15. Each card should show a 4-digit number.
Q1. What is the smallest 4-digit number Asha can use?
L3 ApplySmallest 4-digit: make leading digit as small as possible = 1, then push the rest to end: 1, 0, ?, ?. Need remaining digits sum 14 with last digit largest: 1 + 0 + 5 + 9 = 15 → 1059.
Q2. Analyse: How many 4-digit palindromes have digit sum 14?
L4 Analyse4-digit palindrome form: abba, digit sum = 2a + 2b = 14 → a + b = 7. Pairs (a, b) with a ∈ {1..7}, b ∈ {0..7}: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (7,0). 7 palindromes: 1661, 2552, 3443, 4334, 5225, 6116, 7007.
Q3. Evaluate: Ravi says "The digit sum of a number equals the digit sum of its reversal." Is this always true? Justify.
L5 EvaluateTrue. Reversing a number reorders its digits; it does not add or remove any digits. Since addition is commutative, the sum remains identical.
Q4. Create a 5-digit palindrome whose digit sum is 20 and which is also an even number.
L6 CreateA 5-digit palindrome has form abcba. Sum = 2a + 2b + c = 20. For it to be even, first digit a must be even (since last = first). Try a = 2, b = 5, c = 6: 2 + 5 + 6 + 5 + 2 = 20 ✓ → 25652. Even ✓. Many valid answers.
Assertion–Reason Questions
A: All 3-digit palindromes have the form aba with a ≠ 0.
R: A 3-digit number cannot start with 0.
R: A 3-digit number cannot start with 0.
Answer: (a) — Palindrome means first = last; and first digit cannot be zero for a 3-digit number.
A: The digit sum of 999 is the same as the digit sum of 27.
R: Digit sum depends only on the digits used, not the positions.
R: Digit sum depends only on the digits used, not the positions.
Answer: (a) — 9+9+9 = 27; 2+7 = 9. Wait — 999 → digit sum 27; 27 → digit sum 9. So A is FALSE. Correct answer: (d). R is true, but A is false because 27 ≠ 9's digit sum equals 9.
A: The reverse-and-add process starting at 34 reaches a palindrome in one step.
R: 34 + 43 = 77, which is a palindrome.
R: 34 + 43 = 77, which is a palindrome.
Answer: (a) — R directly explains A.
Frequently Asked Questions
What is a palindrome number?
A palindrome number reads the same forwards and backwards. Examples include 121, 1331, 12321, and 9. In NCERT Class 6 Ganita Prakash Chapter 3, students discover palindromes and explore how to generate them from any starting number.
How do you make a palindrome from any number?
Take any number, reverse its digits, and add the reverse to the original. Repeat with the result if it's not yet a palindrome. For example, 34 + 43 = 77 (palindrome in one step). This reverse-and-add process is explored in NCERT Class 6 Chapter 3.
What is a number line in Class 6 Maths?
A number line is a straight line where numbers are placed at equal intervals. In NCERT Class 6 Ganita Prakash Chapter 3, students use the number line to visualise order, compare numbers, and perform addition and subtraction by jumping along it.
How many 3-digit palindromes are there?
There are 90 three-digit palindromes, from 101 to 999. Each has the form ABA where A is 1-9 (9 choices) and B is 0-9 (10 choices), giving 9 x 10 = 90. Class 6 students explore such counting in Ganita Prakash Chapter 3.
Why are digit patterns important in Number Play?
Digit patterns reveal hidden structure in numbers and help students understand place value, symmetry, and arithmetic properties. Recognising patterns such as palindromes sharpens pattern-recognition skills used throughout NCERT Class 6 mathematics.
Is zero a palindrome?
Yes, 0 is considered a palindrome because it reads the same forwards and backwards. Single-digit numbers 0-9 are all trivially palindromes. NCERT Class 6 Ganita Prakash Chapter 3 treats palindromes starting from multi-digit cases for interesting patterns.
Frequently Asked Questions — Number Play
What is Number Line, Digits and Palindromes in NCERT Class 6 Mathematics?
Number Line, Digits and Palindromes is a key concept covered in NCERT Class 6 Mathematics, Chapter 3: Number Play. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Number Line, Digits and Palindromes step by step?
To solve problems on Number Line, Digits and Palindromes, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 6 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 3: Number Play?
The essential formulas of Chapter 3 (Number Play) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Number Line, Digits and Palindromes important for the Class 6 board exam?
Number Line, Digits and Palindromes is part of the NCERT Class 6 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Number Line, Digits and Palindromes?
Common mistakes in Number Line, Digits and Palindromes include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Number Line, Digits and Palindromes?
End-of-chapter NCERT exercises for Number Line, Digits and Palindromes cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 3, and solve at least one previous-year board paper to consolidate your understanding.
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