The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

In previous classes, we explored numbers in different ways, for instance, rational numbers, whole numbers, integers, multiples, finding factors and more. Here we will deal with some new concepts and tricks to make **playing with numbers** more exciting and fun.

When we write** numbers in general form, **we multiply individual digits with their corresponding place values and then add them together.

- Now let us consider a two-digit number XY, where X and Y are its digits. X is at ten’s place, so we multiply it by 10, and Y is at unit’s place, so it remains the same.
- Therefore, the generalised form of XY = 10X + Y, where ‘X’ can be any digit from 1 to 9, and ‘Y’ can be any digit from 0 to 9.
**For example, the general form of 65 = 10 x 6 + 5.** - Similarly, if we take a three-digit number XYZ, where X is at hundred’s place, we multiply it by 100, Y is at ten’s place, so we multiply it by 10 and Z at unit’s place, so it remains the same. The generalized form of XYZ is 100X + 10Y + Z.
**For example, the general form of 325 = 100 x 3 + 10 x 2 + 5.**

**Reversing the two-digit numbers**

- In case of XY, the original number is 10X + Y, and upon reversal, it becomes 10Y + X. When both numbers are added, the result will always be divisible by 11, and the remainder will be zero.
- Sum of the both the numbers = (10Y + X) + (10X + Y)
- = 10X + Y + 10Y + X
- = 11X +11Y
- = 11(X + Y)
- For example, Adding 85 and 58, we get 143 which is divisible by 11 and the remainder is zero.
- When both the original and reversed numbers are subtracted, the result is always divisible by 9. For example, 85 – 58 =27, which is divisible by nine and the remainder is zero.

**Reversing three-digit Numbers**

- In case of XYZ, the original number is 100X +10Y + Z, and upon reversal, the numbers become 100Z +10Y + X.
- Difference of the both the numbers = (100X +10Y + Z) – (100Z +10Y + X)
- = 100X + 10Y + Z – 100Z – 10Y – X
- = 99X – 0 – 99Z
- = 99(X – Z)
- Consequently, the answer will always be a multiple of 99 and divisible by 11.

**Forming Three-Digit Numbers from Given Three Digits**

- Think of any 3-digit number and form two more different numbers from the same digits. Add these numbers and divide by 37. You will get zero remainder.
- Let us take PQR, then the other numbers would be QRP (the unit’s place digit moved to the left end), and RPQ (the hundred’s digit moved to the right end).
- PQR = 100P + 10Q + R
- QRP = 100Q + 10R + P
- RPQ = 100R + 10P + Q
- = 111 (P + Q + R)
- = 37 × 3 (P + Q + R), divisible by 37.
- For example, 839 + 398 + 983 = 2220/37 = 60, 0 remainder.

- Letters for Digits
- These puzzles are like cracking a code because we substitute some
**letters for digits**. You have to solve the problem to find out which letter represents which digit, keeping some essential rules in mind.

- One letter can represent only one digit.
- The first digit cannot be ZERO, i.e., you cannot write XY as 00XY or 0XY.

- Example 1- Puzzles with Addition
- Let us find the value of X in
**2X8 + 73X = 982**

- In the above addition, 8 + X = 2, that means X = 4.
**Now put 4 in place of X in the equation, 248 + 734 = 982.**

- Example 2- Puzzles with Multiplication
- Let us find the value of X and Y in
**XY x X7 = 129Y**

- Y × 7 = Y, it can be only possible is Y = 0 or Y = 5
- Now for X, if we assume that X = 1 then the maximum value of XY x X7 would be 19 × 17 = 323. Therefore, the value of X cannot be 1. Similarly, If X = 2, then the maximum value of XY x X7 would be less. Therefore, the value of X cannot be 2.
- If X = 3 And Y= 0, XY × X7 = 30 × 37 = 1110.
- If X = 3 And Y = 5, XY × X7 = 35 × 37 = 1295.
- The second possibility yields the correct answer; therefore, X = 3 and Y = 5.

You have studied the rules of divisibility. Here, we shall understand the reasons justifying these rules.

**Divisibility by 2:**It happens if the digit at one’s place is an even number. For example, XY is divisible by 2 only when Y = 0, 2, 4, 6 or 8.**Divisibility by 3:**It happens if the sum of the digits of a number is divisible by 3. For example, in 342, the sum of the digits = 3 + 4 + 2 = 9, which is divisible by 3. Therefore, 342 is divisible by 3.**Divisibility by 5:**It happens if the digit at one’s place is either 0 or 5. Therefore, XY is divisible by 10 only when Y = 0 or Y = 5.**Divisibility by 9:**It happens if the sum of the digits of a number is divisible by 9. For example, in 891, the sum of the digits = 8 + 9 + 1 = 18, which is divisible by 9. Therefore, 891 is divisible by 9.**Divisibility by 10:**It happens if the digit at one’s place is zero. Therefore, XY is divisible by ten only when Y = 0.

- What does a numbers game mean?
- Numbers game uses mathematics to perform some activities where a specific action leads to a greater chance of a particular outcome.
- What do numbers mean in words?
- Numbers in words are written in alphabetical form based on the place value of digits, such as ones, tens, hundreds, and more. For example, 543 is spelt as five hundred forty-three.
- What is the definition of a multiple?
- A multiple is a number obtained by multiplying a whole number with an integer. For example, the multiples of 3 are 3, 6, 9, 12, …
- What does factor mean?
- A factor is a whole number capable of evenly dividing another number. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. Numbers like 5 and 7 are not the factors of 12 because they cannot be divided evenly into 12.
- What are the factors of 18?
- The factors of 18 are 1, 2, 3, 6, 9 and 18.
- Is factor and cause the same?
- No

To explore more, visit MSVgo – a video based learning app designed to explain complex concepts through simple, interactive and explanatory visualisations.

In this chapter, we understood that we could write numbers in general form like a two-digit number PQ = 10 x P + Q. Similarly, we can find the general form 3-digit number, 4-digit number and more.

We also learned that the general form of numbers helps explain **the tests of divisibility** and solving puzzles or** playing with numbers**.

- Rational Numbers
- Linear Equations in One Variable
- Understanding Quadrilaterals
- Practical Geometry
- Data Handling
- Squares and Square Roots
- Cubes and Cube Roots
- Comparing Quantities
- Algebraic Expressions and Identities
- Visualizing Solid Shapes
- Mensuration
- Exponents and Powers
- Direct and Inverse Proportions
- Factorization
- Introduction to Graphs