WORKING WITH NUMBERS

INTRODUCTION TO NUMBERS

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Introduction to Numbers

Introduction to Numbers

Type of Number Description

Counting Numbers {1, 2, 3, ...}

Whole Numbers {0, 1, 2, 3, ...}

Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers p/q : p and q are integers, q is not zero

Irrational Numbers Not Rational

Real Numbers Rational Numbers and Irrational Numbers

Summary of Numbers

WORKING WITH NUMBERS

DIVISIBILITY TEST

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Divisibility Test

DIVISIBILITY BY 2A number is divisible by 2, if its units digit is an even digit i.e. , 2, 4, 6, or 8. DIVISIBILITY BY 3A number is divisible by 3, if the sum of its digits is divisible by 3. DIVISIBILITY BY 4A natural number is divisible by 4, if the number formed by its digits in units and tens places is divisible by 4.Also, if the number formed by its digits in units and tens places is not divisible by 4, then the given number is not divisible by 4.Consider the number 79812. Since 12 is a multiple of 4. So, 79812 is divisible by 4.Consider now the number. Since, 92 is a multiple of 4. So, is divisible by 4. The number is not divisible by 4, because 31 is not a multiple of 4. DIVISIBILITY BY 5A number is divisible by, if its unit digit is 0 or 5. DIVISIBILITY BY 6A number is divisible by, if it is divisible by 2 as well as . 

Divisibility Test

DIVISIBILITY BY 8A natural number is divisible by 8, if the number formed by its digits in units, tens and hundreds places is divisible by 8.Consider the number 79816. Since 816 is a multiple of 8. So, 79816 is divisible by 8. DIVISIBILITY BY 9If the sum of the digits of a natural number is divisible by 9, then the number is divisible by 9.ILLUSTRATION: If is a multiple of 9, where is a digit what is the value of ?SOLUTION: Since is a multiple of 9. Therefore, the sum of its digits is a multiple of 9.. is a multiple of 9 is a multiple of 9 , or 9, or 18, or 27, 36, ... ...(i)But, is a digit. So, can take values, , 9... can take values 8, 9, 10, 11, ..., 17. ...(ii)From (i) and (ii), we get Hence,. DIVISIBILITY BY 11A number is divisible by 11, if the difference of its digits in oddPlaces and the sum of its digits in even places is either or a multiple of 11.Consider three‐digit number 264. For this number, we haveSum of the digits in odd placesSum of the digits in even places So, 264 is divisible by 11.Ex: 61809

Divisibility Test

Example: If is a multiple of 3, where is a digit, what might be the value of?

Example: Without actual division find the remainder when 379843 is divided by 3.

Divisibility Test

Example: If is a multiple of 6, where is a digit, what might be the value of?

Example: If is a multiple of 6, where is a digit, what might be the value of?

Divisibility Test

Example: If is a multiple of 11, where is a digit, what is the value of?

Example: Given that the number is divisible by 11, where is some digit, what are the possible values of ?

Divisibility Test

Example: Given that the number is divisible by 9, where is a digit, what are the possible values of ?

Example: Given that the number is divisible by 4, where is a digit, what are the possible values of a?

WORKING WITH NUMBERS

HCF AND LCM

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HCF and LCM

Finding HCFMethod: (Short cut method) – Most popular method2 |12, 16(Divide by common divisor 2)2 | 6, 8 (Divide by common divisor 2) 3, 4 HCF = 2 x 2 = 4

Method: Continued Division MethodDivide the bigger number by smaller number. Then the divisors are divided in succession by the remainders got. This division should be carried out till we get the remainder zero. The last divisor is the HCF of the given numbers.

HCF and LCM

Example: Find the HCF of 36 and 60.

HCF and LCM

Illustration: Find the LCM of 30 and 12.

Example: Find the HCF and LCM of 10, 15, 20.

HCF and LCM

Example: Find the HCF and LCM of 12, 20, 32.

Example: The HCF of 72 and 252 is 36. Find their LCM.

HCF and LCM

Example: The students of a class can be divided into groups of 6 or groups of 8 without leaving out any student. What will be the minimum number of students in such a class?

Example: A merchant has 120 liters and 180 liters of two kinds of oil. He wants to sell the oil by filling the two kinds of oil in tins of equal volumes. What is the greatest volume of such a tin?