Class 8 Maths Number Play Notes

Chapter 5: Number Play

Overview:
This chapter explores the fascinating patterns, puzzles, and tricks with numbers. Students will learn to identify patterns, explore arithmetic sequences, understand magic squares, and develop reasoning skills with numbers. The focus is on thinking creatively with numbers rather than just performing calculations.

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Key Concepts

1. Number Patterns:
   • Numbers often form patterns, such as sequences: arithmetic or geometric progressions.
   • Example: 2, 4, 8, 16… (geometric); 3, 6, 9, 12… (arithmetic)
2. Magic Squares:
   • A square arrangement of numbers where sum of each row, column, and diagonal is the same.
   • Example: 3×3 magic square with numbers 1–9.
3. Divisibility Tricks and Patterns:
   • Recognizing divisibility by 2, 3, 5, 7, 9, 11 helps in pattern recognition and problem-solving.
   • Example: Numbers divisible by 9 have digits summing to a multiple of 9.
4. Arithmetic Tricks:
   • Sum of consecutive numbers: $1 + 2 + 3 + … + n = \frac{n(n+1)}{2}$1+2+3+…+n=2n(n+1)​
   • Patterns in multiples and squares.
5. Playing with Digits:
   • Reversing digits, forming new numbers, checking sums/differences for patterns.
   • Example: Reversing a 2-digit number and subtracting the smaller from the larger gives a multiple of 9.
6. Puzzles with Numbers:
   • Questions like “find the missing number in the sequence” or “identify the rule in the pattern” help develop logical thinking.

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Important Points to Remember:

• Look for arithmetic relationships in sequences.
• Patterns can involve addition, subtraction, multiplication, or division.
• Some patterns are based on symmetry, reversals, or sums of digits.
• Playing with numbers improves mental calculation and reasoning skills.

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Examples:

1. Sequence Problem:
   • Sequence: 5, 10, 20, 40… Next term = 40 × 2 = 80
2. Magic Square:
   • 3×3 Magic square sum = 15
   8 1 6
3 5 7
4 9 2
   • Check: Row, column, and diagonal sum = 15
3. Digit Trick:
   • 63 → reverse 36 → subtract 63−36 = 27 → divisible by 9

Questions

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A. Very Short Answer Questions (1–10)

1. What is a number pattern?
2. Write the next number: 2, 4, 8, 16, __
3. Write the next number: 5, 10, 15, 20, __
4. Check if 729 is divisible by 9.
5. Check if 572 is divisible by 2.
6. Reverse the number 36 and subtract smaller from larger. What is the result?
7. Write a 3×3 magic square using numbers 1–9.
8. Sum of consecutive numbers from 1 to 10 = ?
9. Write the next number in sequence: 1, 4, 9, 16, __
10. Check whether 154 is divisible by 7.

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B. Short Answer Questions (11–25)

11. Find the next term: 3, 6, 12, 24, __
12. Find the next term: 1, 2, 4, 8, 16, __
13. Find the sum of first 20 natural numbers.
14. Sum of first 15 odd numbers.
15. Find the sum of first 10 even numbers.
16. Check divisibility: 234 by 3.
17. Check divisibility: 815 by 5.
18. Reverse 72, subtract smaller from larger. Result divisible by?
19. Complete the sequence: 7, 14, 28, 56, __
20. Complete the sequence: 10, 20, 30, 40, __
21. Find the missing number: 5, 10, 20, __, 80
22. Check if 1001 is divisible by 7.
23. Find the sum of numbers from 1 to 50.
24. Write the next square number after 36.
25. Find the missing term: 2, 5, 10, 17, __

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C. Word Problems / Application Questions (26–40)

26. A student adds consecutive numbers from 1 to 30. Find the sum.
27. The sum of first n natural numbers = 210. Find n.
28. Find the 10th term of sequence: 7, 14, 21…
29. A number when reversed and subtracted from itself gives 27. Find possible numbers.
30. Fill the blank: 1, 4, 9, 16, 25, __, __
31. A magic square has sum 15. One row = 8,1,6. Verify for column.
32. Find the sum of numbers divisible by 3 from 1 to 30.
33. A number is divisible by 9. Sum of its digits = ?
34. Find next two terms: 2, 6, 12, 20, __, __
35. The numbers 1, 2, 3… form a sequence. Find sum of first 25 numbers.
36. The difference between a number and its reverse = 27. Find one possible number.
37. Check divisibility of 486 by 9.
38. Complete the pattern: 5, 11, 17, 23, __, __
39. A magic square uses numbers 1–9. Check if diagonals sum = 15.
40. Find the next term in sequence: 1, 8, 27, 64, __

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D. Higher Order / Thinking Questions (41–50)

41. Find the sum of first 50 odd numbers.
42. Find the sum of first 50 even numbers.
43. Create a 3×3 magic square whose sum = 15.
44. Reverse 81 and subtract smaller from larger. Result divisible by?
45. Write the next three terms: 4, 9, 16, 25, __, __, __
46. A number is divisible by 3 and 9. Write one example.
47. The difference between 72 and its reverse is divisible by which number?
48. Find the 15th term of the sequence: 3, 6, 9, 12…
49. A number is divisible by 2, 3, 5. Find the smallest number.
50. Write a sequence of 5 numbers whose sum = 50 and each number increases by 2.

Answers

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A. Very Short Answer Questions (1–10)

1. Number pattern: A sequence of numbers arranged according to a rule or formula.
2. 2, 4, 8, 16 → Next = 32 (each term ×2)
3. 5, 10, 15, 20 → Next = 25 (each term +5)
4. 729 → sum of digits = 7+2+9=18 → divisible by 9 → Yes
5. 572 → last digit = 2 → divisible by 2 → Yes
6. 36 → reverse = 63 → 63−36=27
7. 3×3 Magic square (sum = 15): 8 1 6
3 5 7
4 9 2
8. Sum 1+2+…+10 = 10×11/2 = 55
9. Sequence: 1, 4, 9, 16 → next = 25 (perfect squares: 1²,2²,3²,4²…)
10. 154 ÷ 7 = 22 → Yes divisible by 7

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B. Short Answer Questions (11–25)

11. Sequence: 3,6,12,24 → next = 48 (×2)
12. Sequence: 1,2,4,8,16 → next = 32 (×2)
13. Sum of first 20 natural numbers = 20×21/2 = 210
14. Sum of first 15 odd numbers = 15² = 225
15. Sum of first 10 even numbers = 10×(10+1) = 10×11 = 110 → Wait, check: sum of first n even numbers = n(n+1) → 10×11=110 ✅
16. 234 → sum digits 2+3+4=9 → divisible by 3 → Yes
17. 815 → last digit 5 → divisible by 5 → Yes
18. 72 → reverse 27 → 72−27=45 → divisible by 9 → Yes
19. 7,14,28,56 → next = 112 (×2)
20. 10,20,30,40 → next = 50 (+10)
21. 5,10,20,__ ,80 → pattern ×2 → missing = 40
22. 1001 ÷ 7 = 143 → Yes
23. Sum 1+2+…+50 = 50×51/2 = 1275
24. Next square after 36 = 49 (7²)
25. 2,5,10,17 → pattern: n²+1 → next = 4²+1=17 (check) → next = 5²+1=26 ✅

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C. Word Problems / Applications (26–40)

26. Sum 1+2+…+30 = 30×31/2 = 465
27. Sum = n(n+1)/2 = 210 → n(n+1)=420 → n=20 ✅
28. 10th term: 7+ (10−1)×7=7+63= 70
29. Number − reverse = 27 → e.g., 63−36=27 ✅
30. Sequence 1,4,9,16,25 → next =36, then 49
31. Magic square row = 8+1+6=15 → check column 8+3+4=15 ✅
32. Numbers divisible by 3 from 1–30: 3,6,9,…,30 → sum=165
33. Number divisible by 9 → sum digits multiple of 9 ✅
34. 2,6,12,20 → differences 4,6,8 → next differences 10 → 20+10=30, next diff 12 → 30+12=42
35. Sum 1+2+…+25 = 25×26/2 = 325
36. Difference =27 → e.g., 63−36=27
37. 486 sum digits 4+8+6=18 → divisible by 9 → Yes
38. 5,11,17,23 → difference =6 → next =29, 35
39. Magic square numbers 1–9 → check diagonals sum = 15 ✅
40. 1,8,27,64 → cubes 1³,2³,3³,4³ → next = 125 (5³)

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D. Higher Order / Thinking Questions (41–50)

41. Sum first 50 odd numbers = 50² = 2500
42. Sum first 50 even numbers = 50×(50+1)=50×51=2550
43. 3×3 magic square sum 15:

8 1 6
3 5 7
4 9 2

44. 81 → reverse 18 → 81−18=63 → divisible by 9 → Yes
45. 4,9,16,25 → next =36,49,64
46. Number divisible by 3 and 9 → e.g., 18
47. Difference 72−27=45 → divisible by 9 → Yes
48. 15th term of 3,6,9,12,… = 3+(15−1)×3 = 3+42=45
49. Number divisible by 2,3,5 → LCM(2,3,5)=30 → smallest = 30
50. Sequence sum=50, difference 2 → sequence: 8,10,12,14,6 → sum=50 ✅ (one example)