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1. Introduction

Numbers are not just tools for counting; they hide many interesting patterns and secrets. By writing numbers in their expanded form and studying the letters and digits, we can discover fascinating properties of numbers. For example, why is a number divisible by 9 when the sum of its digits is divisible by 9? Why is a number divisible by 11 when the alternating sum of its digits is a multiple of 11? These questions belong to the wonderful branch of mathematics called number theory, and they are studied in this chapter through the concept of playing with numbers.

In this chapter we will learn how to write numbers in generalised (expanded) form, how to solve puzzles involving numbers in which some digits are replaced by letters, and the important divisibility tests for numbers 2, 3, 4, 5, 6, 8, 9, 10 and 11. Understanding these tests helps us check large numbers quickly without actually dividing them.

2. Numbers in Generalised Form

Any two-digit number ab (where a is the tens digit and b is the units digit) can be written in its generalised (expanded) form as: $$ab = 10a + b$$

For example, 47 = 10 x 4 + 7. Similarly, a three-digit number abc can be written as: $$abc = 100a + 10b + c$$

For example, 523 = 100 x 5 + 10 x 2 + 3.

Reversing Two-Digit Numbers

If a two-digit number is ab = 10a + b, then its reverse is ba = 10b + a. - The sum of the number and its reverse is 10a + b + 10b + a = 11a + 11b = 11(a + b), which is always divisible by 11. - The difference between the number and its reverse is (10a + b) - (10b + a) = 9a - 9b = 9(a - b), which is always divisible by 9.

3. Puzzles with Letters

In such puzzles, some letters stand for digits of numbers. We use the properties of numbers and divisibility to find the values of the letters. Each letter represents a unique digit.

Example

Solve the puzzle: A B + A B

B C C

Working through the addition: B + B gives a units digit C, and carrying may occur. Using the divisibility and place value rules, we can determine A, B and C uniquely. Such puzzles test our understanding of place value and arithmetic.

4. Tests of Divisibility

Divisibility by 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6 or 8).

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5.

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Divisibility by 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 10

A number is divisible by 10 if its last digit is 0.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11.

5. The Magic of Number Patterns

Multiples of 9

The sum of the digits of a multiple of 9 is always 9 or a multiple of 9. For example, 9, 18, 27, ... 9 x 12 = 108 and 1 + 0 + 8 = 9.

Reversible Numbers

Numbers like 12 and 21 are reverses of each other. Their sum 12 + 21 = 33 is divisible by 11, and their difference 21 - 12 = 9 is divisible by 9, as shown by the generalised form.

These patterns show that numbers are not random but follow beautiful mathematical laws.

Quick Revision Tables

Table 1: Divisibility Tests

Divisibility Test
By 2 Last digit is even
By 3 Sum of digits is divisible by 3
By 4 Last two digits form a number divisible by 4
By 5 Last digit is 0 or 5
By 6 Divisible by both 2 and 3
By 8 Last three digits form a number divisible by 8
By 9 Sum of digits is divisible by 9
By 10 Last digit is 0
By 11 Difference of sums of digits at odd and even places is 0 or a multiple of 11

Table 2: Generalised Form

Number Generalised form
ab (two digit) 10a + b
abc (three digit) 100a + 10b + c
ab + ba 11(a + b), divisible by 11
ab - ba 9(a - b), divisible by 9

Mind Map

graph TD A["Playing with Numbers"] --> B["Generalised form"] B --> C["ab = 10a + b"] B --> D["abc = 100a + 10b + c"] A --> E["Reversing numbers"] E --> F["ab + ba = 11(a + b), divisible by 11"] E --> G["ab - ba = 9(a - b), divisible by 9"] A --> H["Puzzles with letters"] H --> I["Use place value and divisibility"] A --> J["Divisibility tests"] J --> K["2, 3, 4, 5, 6, 8, 9, 10, 11"]

Important Diagrams (SVG)

Diagram 1: Sum and Difference of a Number and its Reverse

Number and its Reverse ab = 10a + b example: 47 = 40 + 7 ba = 10b + a example: 74 = 70 + 4 Sum = ab + ba = 11(a + b), divisible by 11 Difference = ab - ba = 9(a - b), divisible by 9 Check with 47 and 74 47 + 74 = 121 = 11 x 11, divisible by 11 74 - 47 = 27 = 9 x 3, divisible by 9 Golden Rule: The sum of a number and its reverse is divisible by 11; their difference is divisible by 9.

Diagram 2: Divisibility Test for 11

Is 1331 divisible by 11? 1 3 3 1 1 3 3 1 odd place 1 odd place 3 even place 3 even place 1 Sum of odd places 1 + 3 = 4 Sum of even places 3 + 1 = 4 4 - 4 = 0, so divisible by 11 Golden Rule: For divisibility by 11, the odd-place and even-place digit sums must differ by 0 or a multiple of 11.

Common Mistakes

  1. Students forget that the digit at the tens place in a two-digit number is multiplied by 10 and the hundreds digit by 100 while writing the generalised form.
  2. While reversing a number, students reverse the digits but forget to reassign the place values correctly.
  3. Students confuse the divisibility tests of 3 and 9. Both use the sum of digits, but 3 needs the sum divisible by 3 while 9 needs it divisible by 9.
  4. In divisibility by 4 and 8, students check the whole number instead of the last two digits (for 4) or the last three digits (for 8).
  5. In the divisibility test for 11, students mix up which places are odd and which are even, counting from the wrong end.
  6. Students think a number divisible by 6 must have its last digit 6. Actually it must be divisible by both 2 and 3, so it must be even and its digit sum divisible by 3.
  7. In puzzles, students guess the digits randomly instead of using place value and carrying rules systematically.

Exam Tips

  1. Write the generalised form first in every puzzle and divisibility proof, as it is the key to the solution.
  2. Memorise the divisibility tests and practise applying them to large numbers quickly.
  3. For divisibility by 11, carefully count places from the left (or right, consistently) and compute both sums.
  4. In puzzles, start from the units column and use the carrying rules step by step.
  5. Use the result that ab + ba is divisible by 11 and ab - ba is divisible by 9 in short-answer questions.
  6. Always verify divisibility by actually dividing, especially in multiple-choice questions.
  7. Revise the pattern that the sum of digits of any multiple of 9 is itself a multiple of 9.

Conclusion

Playing with numbers reveals the hidden beauty and structure of the number system. In this chapter we learnt how to express numbers in their generalised form using place values, how to reverse numbers and the interesting results that follow - the sum of a number and its reverse is divisible by 11, while their difference is divisible by 9. We solved puzzles in which digits are replaced by letters, and we studied the divisibility tests for numbers 2 through 11. These ideas build our number sense and lay the groundwork for prime factorisation, HCF, LCM and cryptography in higher classes. Most importantly, this chapter shows that mathematics is not just about rules but about discovering fascinating patterns.