CBSE Notes for Class 6 Maths Chapter 2 Playing With Numbers Summary is highly helpful for students who want to understand all the exercise questions. NCERT Class 6 Maths Notes and Practice Problems have been reviewed by our best Mathematics teachers. We have provided detailed notes and all the solutions for NCERT Maths class 6 Playing With Numbers. so that you can solve the questions in class 6 mathematics textbook seamlessly. CBSE worksheets for class 6 Maths are available at free of cost to all the students.

**Board:** Central of Secondary Education

**Class:** Class 6

**Subject:** Maths

**Chapter Name:** Playing With Numbers.

**Factors and Multiples:**

**Factor:** A factor of a number is an exact divisor of that number.

In other words, a factor of a number is that number which completely divides the number without leaving a remainder.

Each of the numbers 1, 2, 3, 4, 6 and 12 is a factor of 12. However, none of the numbers 5, 7, 8, 9, 10 and 11 is a factor of 12.

**Multiple:** A multiple of a number is a number obtained by multiplying it by a natural number.

If we multiply 3 by 1, 2, 3, 4, 5, 6, , we get

3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12, 3 × 5 = 15, 3 × 6 = 18,

Thus, 3, 6, 9, 12, 15, 18, are multiples of 3.

Every multiple is equal to or greater than the given number.

**Perfect Number:**

Factors of 6 are 1, 2, 3 and 6

Now, the sum of the factors of 6

= 1 + 2 + 3 + 6 = 12 = 2 times of 6

Factors of 28 are 1, 2, 4, 7, 14 and 28

Now, the sum of the factors of 28 = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 times 28

The numbers like 6 and 28 are called perfect numbers.

A number is called a perfect number if the sum of all its factors is equal to twice the number.

12 is not a perfect number.

Factors of 12 are: 1, 2, 3, 4, 6, 12

Now the sum of factors of 12 = 1 + 2 + 3 + 4 + 6 + 12 = 28 is not equal to 2 12 = 24

**Even Numbers:** All multiples of 2 are called even numbers.

We know that 2, 4, 6, 8, 12, 14, are multiples of 2.

Hence, 2, 4, 6, 8, 10, 12, 14, are even numbers.

Clearly, a number is even if is divisible of 2 or 2 is a factor of it.

**Odd Numbers:** Numbers which are not multiples of 2 are called odd numbers.

Clearly, 1, 3, 5, 7, 9, 11, 13, 15, are odd numbers.

Also, a number is either even or odd. A number cannot be both even as well as odd.

In order to list all factors of a number. We may follow the following procedure.

**Prime and Composite Numbers:**

**Prime numbers:** Numbers with only two factors, 1 and the number itself, are known as prime numbers. Examples are 2, 3, 5, 7, 11, 13,

**Composite Numbers:** Numbers with more than two factors are called composite numbers.

Examples are 4, 6, 8, 9, 10, 12,

Number 1 is neither prime nor composite.

TEST OF DIVISIBILITY |
||

No. |
Divisibility Test |
Examples |

2 | Unit digit should be 0 or even | 4096, 23548 as they end with 6 and 8 i.e., even numbers |

3 | The sum of digits of no. should be divisible by 3. | 2143251, sum of the digits is 18 and it is divisible by 3 |

4 | The no formed by last 2 digits of given no. should be divisible by 4. | 548, here 48 ÷ 4 = 12 and it is divisible by 4 |

5 | Unit digit should be 0 or 5. | 4095 and 235060 as they have 5, 0 at unit places. |

6 | No should be divisible by 2 & 3 both. | 753618, sum of the digits is 30 and it is divisible by 2 and 3. |

8 | The number formed by last 3 digits of given no. should be divisible by 8. | 5432, here 432 is divisible by 8 |

9 | Sum of digits of given no, should be divisible by 9. | 125847, sum of the digits is 27 and it is divisible by 9 |

11 | The difference between sums of the digits at even & at odd places should be zero or multiple of 11. | 9582540, here sum of odd places- sum of even places (22 – 11 = 11) and 11 is a divisible by 11 |

25 | Last 2 digits of the number should be. 00, 25, 50 or 75. | 2500, 2550 etc |

**H.C.F. & L.C.M.**

**H.C.F. (Highest Common Factor):
**The greatest number which divides all the given numbers is called Highest Common Factor (H.C.F.). e.g., 18 and 30 are the given numbers 6 is the only greatest number which divides both 18 and 30 exactly

**NOTE :-**

The product of two numbers a and b is equal to the product of their L.C.M. and H.C.F.

a × b = H.C.F. × L.C.M.

Product of ‘n’ number = (H.C.F. of each pair)^{n – 1} × L.C.M. of n pair

**L.C.M. (Least Common Multiple):
**The least number which is exactly divisible by all the given numbers is Least Common Multiple 24 is only least common multiple of 6, 8 and 12

**Ex.**L.C.M. of 6, 8 and 12 is 24

**Solved Examples**

**Problem 1.**

Find the least number which when divided by 20, 25, 35 and 40 leaves remainder 14, 19, 29 and 34 respectively

**Sol.**

(20 – 14) = 6, (25 – 19) = 6, 35 – 29 = 6 40 – 34 = 6= r

Required number = L.C.M. of (20, 25, 35 and 40) – 6 = 1400 – 6 = 1394

**Problem 2.
**Find the least number which when divided by a, b and c leaves the same remainder ‘r’ in each case

**Sol:**

Let L.C.M. of a, b and c = M

Required number = M + r

**Problem 3.
**The traffic lights at three different road crossing change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at 8 : 20 : 00 hours, then at what time will they again change simultaneously?

**Sol.**

Interval of change = (L.C.M. of 48, 72, 108) sec = 432 sec

So the light will again change simultaneously

after every 432 seconds i.e., 7min 12 sec.

Hence next simultaneous change will take place at 6 : 27 : 12 hrs.

**Problem 4.
**Find the greatest number that will divide 148, 246, 623 leaving remainders 4, 6 and 11 respectively

**Sol.**

Required No. = H.C.F. of (148 – 4), (246 – 6) and (623 – 11)

= H.C.F. of (144, 240 and 612)

H.C.F. = 12

Required No. =12

**H.C.F & L.C.M of Fractions
**

**Ex.**

L.C.M. of two distinct natural numbers is 211, what is their H.C.F.?

**Sol.**

211 is a prime number, so there is only one pair of distinct numbers possible whose L.C.M. is 211, i.e., 1 and 211, H.C.F. of 1 and 211 is 1.

**Ex.
**Find number of prime factors in 2

^{222}× 3

^{333}× 5

^{555 }

**Sol.**

No. of prime factors = 222 + 333 + 555 = 1110

**Multiple Choice Questions**

**Problem 1.**

Which of the following numbers is a perfect number?

(A) 4

(B) 12

(C) 8

(D) 6

**Problem 2.
** Which of the following are not twin-primes?

(A) 3, 5

(B) 5, 7

(C) 11, 13

(D) 17, 23

**Problem 3.
**Which of the following are co-primes?

(A) 8, 10

(B) 9, 10

(C) 6, 8

(D) 15, 18

**Problem 4.
**Which of the following is a prime number?

(A) 263

(B) 361

(C) 323

(D) 324

**Problem 5.
**The number of primes between 90 and 100 is

(A) 0

(B) 1

(C) 2

(D) 3

**Problem 6.
**Which of the following numbers is a perfect number?

(A) 16

(B) 8

(C) 24

(D) 28

**Problem 7.
**Which of the following is a prime number?

(A) 203

(B) 139

(C) 115

(D) 161

**Problem 8.
**The total number of even prime numbers is

(A) 0

(B) 1

(C) 2

(D) unlimited

**Problem 9.
**Which one of the following is a prime number?

(A) 161

(B) 221

(C) 373

(D) 437

**Problem 10.
**The least prime is

(A) 1

(B) 2

(C) 3

(D) 5

**Problem 11.
**Which one of the following number is divisible by 3?

(A) 27326

(B) 42356

(C)73545

(D) 45326

**Problem 12.
**Which of the following numbers is divisible by 4?

(A) 8675231

(B)9843212

(C) 1234567

(D) 543123

**Problem 13.
**Which of the following numbers are divisible by 6?

(A) 672

(B) 813

(C) 7312

(D) 1236

(E) 4314

(F) 689

(G) 263

(I) 8135

(J) 7236

(H) 164

**Problem 14.
**From the following, find the numbers divisible by 8.

(A) 328

(B) 4728

(C) 8256

(D) 9096

(E) 6324

(F) 8004

(G) 5368

(H) 6072

(I) 4568

(J) 4821

**Problem 15.
**From the following, find the numbers divisible by 9.

(A) 8163

(B) 7214

(C) 8353

(D) 6345

(E) 1584

(F) 3617

(G) 6273

(H) 8001

(I) 4375

(J) 8931

**Problem 16.
**Identify the numbers divisible by 10.

(A) 29

(B) 430

(C) 89

(D) 77

(E) 120

(F) 33

(G) 17908

(H) 3640

**Problem 17.
**Identify the numbers divisible by 11.

(A) 71412

(B) 376277

(C) 6116

(D) 86124

(E) 643214

(F) 20438

(G) 48295

(H) 14909

(I) 97526

(J) 563761

**Problem 18.
**The HCF of two consecutive odd numbers is

(A) 1

(B) 2

(C) 0

(D) non-existant

**Problem 19.
**The HCF of an even number and an odd number is

(A) 1

(B) 2

(C) 0

(D) non-existant

**Problem 20.**

The LCM of 24, 36 and 40 is

(A) 4

(B) 90

(C) 360

(D) 720

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