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 2222 × 3333 × 5555
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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