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**NCERT / CBSE NOTES : Chapter Summary**

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__Playing with Numbers__

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**General Form of Numbers**

Various types of numbers such as Natural numbers, Whole numbers, Integers, Rational numbers and the various properties such as closure, associative, commutative and distributive.

2-digit number

The number in the general form can be written as for example 26 = 2 x 10 + 6. General Form of a 2-digit Number 10 × a + 1 × b

- The sum of a 2-digit number and the number obtained by interchanging its digits is always divisible by 11.
- The difference between a 2-digit number and the number obtained by interchanging its digits is always divisible by 9.

Assume ab is a 2-digit number.

• a is the tens digit

• b is the ones digit

ab = 10 × a + 1 × b.

3-digit number

General form of a 3-digit number is 100 × a + 10 × b + 1 × c.

- The difference between a 3-digit number and a number obtained by reversing its digits is always divisible by 99.

Assume abc is a 3-digit number, where:

• a is the hundreds digit

• b is the tens digit

• c is the ones digit

abc = 100 × a + 10 × b + 1 × c

Letters for digits

Assume that each letter in a puzzle stands for just one digit and each digit is represented by a one letter, so it is like cracking a code. Here problems of addition and multiplication to solving puzzles.

For example, eighty one will be written as 81 not as 081 or 0081.

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**Tests of Divisibility**

The tests of divisibility with 10, 5, 2, 3, 6, 4, 8, 9 and 11. In this, learn why the numbers are divisible by 10, 5, 2, 3, 6, 4, 8, 9 and 11.

A number is said to be divisible by another number, when the remainder is zero.

• A number is divisible by 10, if its ones digit is 0.

• A number is divisible by 5, if its ones digit is 0 or 5.

• A number is divisible by 2, if its ones digit is 0, 2, 4, 6 or 8.

• If a number is divisible by 10, then the number is also divisible by 2 and 5.

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

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

• If a number is divisible by 9, then the number is also divisible by 3.

• If a number is divisible by 3, then it may not necessarily be divisible by 9.

• If a number is divisible by 6, then the number is divisible by 2 as well as 3.

• If a number is divisible by 11, then the difference of its digits in odd places and the sum of its digits in

even places is either 0 or a multiple of 11.

• If a number is divisible by 4, then the number formed by its digits in units and tens places is divisible by 4.