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**Points to Remember**** **

â€¢ A two digit number can be written in generalized form as 10

â€¢ A three digit number can be written in generalized form as 100

â€¢ Generalized form of numbers are helpful in solving puzzles or number games.

â€¢ A number is divisible by 10, if its ones digits is 0.

â€¢ A number is divisible by 5, if its ones digits is 0 or 5.

â€¢ A number is divisible by 2, if its ones digits is 0, 2, 4, 6 or 8.

â€¢ A number is divisible by 9, if the sum of its digits is divisible by 9.

â€¢ A number is divisible by 3, if the sum of its digits is divisible by 3.**We Know That**

Numbers are of various types such as natural numbers, whole numbers integers, fractional and rational numbers. They are full of fun and magic. We also know about the various divisibility tests of numbers. We can enjoy, the magic and wonder of numbers For example,

9 x 1 =

9 x 2 = 18 â†’ 1 + 8 =

9 x 3 = 27 â†’ 2 + 7 =

9 x 123 = 1107 â†’ 1 + 1 + 0 + 7 =

1234 x 9 = 11106 â†’ 1 + 1 + 1 + 0 + 6 =

12345 x 9 = 111105 â†’ 1 + 1 + 1 + 1 + 0 + 5 =

123456 x 9 = 1111104 â†’ 1 + 1 + 1 + 1 + 1 + 0 + 4 = **NUMBERS IN GENERAL FORM**

We can express a number in general form using place value system, we may call it the expanded form of the number. Let is a two digit number. We can write it in generalized form as Similarly, a 3-digit number can be written as

**Games with numbers**

Reversing the digits (2-digit number)**Example:** Sudaram considered any number of 2-digits.

Writing the given number in generalized form:

Reversing the digits and writing the new number in generalized form:

Adding, we get

i.e. he got

11 x [Sum of the digits of the chosen number]**Reversing The Digits and Subtracting**

Let Sundaram chooses a number

Generalised form =

Reversing the digits, new number

Generalised form =

Subtracting

(when a > b)

[Difference of the digits of the chosen number]

or

(when b > a)

[Difference of the digits of the chosen number]**Reversing The Digits** **of A 3-Digit** **number**

Note:When we reverse the digits of a 3-digit number then the middle digit (i.e. tens digit) remains unchanged.

**Letters for Digit **

Note:For problems of addition and multiplication, we follow the following rules while during various puzzles.

(i) Each letter in a puzzle must stand for just one digit and each must be represented by just one

letter.

(ii) The first digit of a number cannot be zero.

**Example: **Find A and B such that**Solution:** We have to choose B such that B x 3

It is possible for B = 0 or B = 5

Now, let us look for A,

If

A = 1, then AB x AB

10 x 13 = 130 For B = 0

or

15 x 13 = 195 For B = 5

which is less than 570 or 575

If A = 3, then

30 x 33 = 990 For B = 0

and

35 x 33 = 1155 For B = 5

which is greater than 570 or 575

If A = 2, then

20 x 23 = 460 For B = 0

and

25 x 23 = 575 For B = 5

The first possibility (20 x 23) fails but

The second one is correct

âˆ´ The required values of A and B are

A = 2 and B = 5 **Solve Examples:Question 1. On multiplying 121 and its reverse, we geta. 14641b. 14541c. 14441d. None of the above.Solution: **A is the correct option. The reverse of 121 is 121, hence 121 Ã— 121 = 14641.

a. 2 Ã— 100 + 3 Ã— 10 + 7

b. 2 Ã— 10 + 3Ã— 10 + 7

c. 2 Ã— 100 + 2 Ã— 100 + 7

d. 2 Ã— 100 + 3 Ã— 10 + 7

Solution:

abc = aÃ— 100 + b Ã— 10+ c

So, 237 = 2 Ã— 100 + 3 Ã— 10 + 7

So only one expression is in general form.

a. 1, 3 0r 5

b. 1, 3, 7 or 9

c. 7 or 9

d. 1 or 7

Solution:

a. 4683

b. 7321

c. 1428

d. 5631

Solution:

a. 18, 6, 9

b. 18, 36, 6

c. 36, 54, 72

d. None

Solution:

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