06.03 Maths
Ch. 03. Playing With Numbers
लेखक
ॐ जितेन्द्र सिंह तोमर
M.A., B.Ed., DNYS, MASSCOM.
10/4/10/2/2022
3.1 Multiples and Factors and
a) Multiples
Multiples means a Table.
* Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
* Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, ...
* Multiples of 6 are 6, 12, 18, 24, 30, ...
* every multiple of a number is greater than or equal to that number.
* the number of multiples of a given number is infinite.
* every number is a multiple of itself.
b) Factors:- A factor of a number is an exact divisor of that number.
* The factors of 4 are 1, 2 and 4.
* The factors of 18 are 1, 2, 3, 6, 9 and 18
* A number is a multiple of each of its factors.
* 1 is a factor of every number.
* every number is a factor of itself.
* every factor of a number is an exact divisor of that number.
* every factor is less than or equal to the given number.
* number of factors of a given number are finite.
Perfect Number
A number for which sum of all its factors is equal to twice the number is called a perfect number.
Factors of 6 = 1, 2, 3 and 6
1+2+3+6 = 12
= 2 × 6
Factors of 28 = 1, 2, 4, 7, 14 and 28.
1 + 2 + 4 + 7 + 14 + 28 = 56
= 2 × 28.
3.2 Prime and Composite Numbers
Prime Numbers
The numbers which has only two factors 1 and itself are called Prime numbers.
As 2, 3, 5, 7, 11, 13 .......
Co-Prime Number
Two numbers having only 1as a common factor are called Co-Prime Number.
(4, 15), (7, 15), (12, 49), (18, 23) etc
Composite Numbers
Numbers having more than two factors are called Composite numbers.
As 4, 6, 8, 9, 10, 12.....
*1 is neither a prime nor a composite number.
Even And Odd Numbers
Even Numbers
The number divisible by 2 is called even numbers. Numbers 2, 4, 6, 8, 10, 12, 14, ...?
* Every even number is a multiple of 2.
Odd Numbers
The number which are not divisible by 2 or multiple of 2 is called odd numbers.
Numbers 1, 3, 5, 7, 9, 11, ... are called odd numbers.
Sieve of Eratosthenes
Greek Mathematician Eratosthenes, in the third century B.C. use the method. to find all prime numbers from 1 to 100, as shown below.
Step 1 : Cross out 1 because it is not a prime number.
Step 2 : Encircle 2, cross out all the multiples of 2, other than 2 itself, i.e. 4, 6, 8 and so on.
Step 3 : You will find that the next uncrossed number is 3. Encircle 3 and cross out all the multiples of 3, other than 3 itself.
Step 4 : The next uncrossed number is 5. Encircle 5 and cross out all the multiples of 5 other than 5 itself.
Step 5 : Continue this process till all the numbers in the list are either encircled or crossed out.
All the encircled numbers are prime
numbers. All the crossed out numbers, other than 1 are composite numbers. This method is called the Sieve of Eratosthenes.
* 2 is the smallest prime number which is even.
* Every prime number except 2 is odd.
Tests for Divisibility of Numbers
Divisibility by 10
* If a number has 0 in the ones place then it is divisible by 10.
10, 50, 400, 7650 50000
Divisibility by 5
A number which has either 0 or 5 in its ones place is divisible by 5
10, 35, 50, 405, 7650, 50010
Divisibility by 2
A number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its ones place.
10, 32, 54, 408, 7650, 50012
Divisibility by 3
If the sum of the digits is a multiple of 3, then the number is divisible by 3.
For 21
2+1=3, Sum 3 is multiple of 3, so 21 is divisible by 3
For 25
2+5=7, Sum 7 is not multiple of 3, so 25 is not divisible by 3
For 36
3+6=9, Sum 9 is multiple of 3, so 36 is divisible by 3
For 37
3+7=10, Sum 10 is not multiple of 3, so 37 is not divisible by 3
For 54
5+4=9, Sum 9 is multiple of 3, so 54 is divisible by 3
For 219
2+1+9=12. Sum 12 is multiple of 3, so 219 is divisible by 3
For 260
2+6+0 = 8. Sum 8 is not multiple of 3, so 260 is not divisible by 3
Divisibility by 6
If a number is divisible by 2 and 3 both then it is divisible by 6 also.
Divisibility by 9
if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.
For 4608
4 + 6 + 0 + 8 = 18,
Sum 18 is multiple of 9, so 4608 is divisible by 9
For 5283
5 + 2 + 8 + 3 = 18
Sum 18 is multiple of 9, so 5283 is divisible by 9
For 5021823
5 + 0 + 2 + 1 + 8 + 2 + 3 = 21
Sum 21 is not multiple of 9, so 5021823 is not divisible by 9
Divisibility by 4
A number with 3 or more digits is divisible by 4, if the number formed by the last two digits (T-O) is divisible by 4.
Divisibility by 8
A number with 4 or more digits is divisible by 8, if the number formed by the last three digits (H-T-O) is divisible by 8.
Divisibility by 11
If the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number.
If the difference is either 0 or divisible by 11 [or multiple of 11], then the number is divisible by 11.
[odd sum – even sum = 0 or multiple of 11]
If a number is divisible by another number then it is divisible by each of the factors of that number.
Co-prime Numbers
Two numbers having only 1 as a common factor are called co-prime
numbers.
Thus, 4 and 15 are co-prime numbers, 7 and 15, 12 and 49, 18 and 23 co-prime numbers.
Highest Common Factor (HCF)
or
Greatest Common Divisor (GCD)
The Highest (or greatest) Common Factor of two or more given numbers is the Highest Common Factors (HCF) or Greatest Common Divisor (GCD) of their given number.
Lowest Common Multiple (LCM)
The Lowest Common Multiple (LCM) of two or more given numbers is
the lowest (or smallest or least) of their common multiples.
* Calculate HCF for maximum capacity or calculate LCM for minimum capacity
Chapter-3
Playing with numbers
Factors of a number:
A factor of a number is an exact divisor of that number. For eg. 1.2.3, 6, 9 and 18 are factors of 18.
Perfect number:
The number which is equal to the sum of its proper factors.
For e.g. 6 is a perfect number, as proper factors of 6 are: 1, 2, 3 and 6-1+2+3.
Prime numbers:
A prime number is a natural number greater than I which has exactly two factors, 1 and itself.
Composite numbers: Numbers with more than two factors.
1 is neither prime nor composite (as it has only one factor)
Co-primes: Number with only 1 as a common factor. For e.g. 9 and 14 are co-primes.
Twin primes: Two consecutive prime numbers, having a difference of 2.
eg. 3 and 5 are twin primes.
Prime triplets: Three consecutive prime numbers, having a difference of 2.
e.g. 3,5,7
Even number: Natural number that is exactly divisible by 2. For eg. 22, 456
Odd numbers: Natural numbers that cannot be divided by 2 exactly e.g. 13,177
Prime factorization: Prime factorization is the method of finding the factors of a number that are all prime. Prime factorization of any number is unique.
Factor tree: Factor tree is a tool that breaks down any number into its prime factors. For e.g. Factor tree of 48
H.C.F. (Highest Common Factor): HCF of two or more numbers is the highest common factor that divides all the given numbers exactly.
LC.M. (Least Common Multiple): LCM of two or more numbers is the smallest/least number that is exactly divisible by the numbers.
Product of two numbers=HCF × LCM
Divisibility rules:
A number is divisible by :
Number
Divisibility rule
2 –> If last digit is 0, 2, 4, 6 or 8
3 –> If the sum of digits is divisible by 3
4 –> If the last two digits are divisible by 4
5 –> If the last digit is 0 or 5
6 –> If the number is divisible by 2 and 3
8 –> If last 3 digits are divisible by 8
9 –> If the sum of the digits is divisible by 9
10 –> If the last digit is 0
11 –> Subtract the last digit from the number formed by the remaining digits. If new number is divisible by 11, the original number is divisible by 11.
Questions:
1. Which is the smallest odd prime number?
2. How many prime numbers are there between1 to 50?
3. Find all pairs of prime numbers less than 15, whose difference is 2.
4. How many twin primes are there between 20 and 30?
5. Find the greatest 4 digit number divisible by 4.
6. Find the greatest 2 digit prime number.
7. Determine first five multiples of 12.
8. Find the HCF of 9 and 13.
9. Find the LCM of 12 and 15.
10. A vessel has 10 litres 500 ml of milk. In how many glasses each of capacity 150 ml can it be filled?
11. Find all numbers less than 60, which are common multiples of 2 and 3.
12. Find all prime numbers less than 15.
13. Determine all the factors of 27.
14. Find the missing number :
15. Complete the factor tree:
16. Find the least number which when divided by 12, 16, 24 and 36 leaves the remainder 7 in each case.
17. Three persons start walking at the same time. The lengths of their steps be 80cm, 85cm and 90cm respectively. Find the minimum distance that can be measured in exact number of steps.
18. Find the sum of prime numbers between 1 and 10.
19. A machine, on an average, manufactures 250 screws a day. How many screws did it manufacture in the month of November?
20. The Length, breadth and height of a room are 825cm, 675cm and 450 cm respectively. Find the maximum length of tape that can measure all the dimensions exact number of times.
21. Determine first five multiples of 9.
22. Find the HCF of 24 and 36.
23. Find the common factors of 4, 8 and 12.
24. Find the smallest 4-digit number which is divisible by 6, 8 and 9.
25. Find the LCM of 12, 15 and 45.
26. Find the HCF of 27 and 63.
27. If a number is divisible by 10, then what is its one's digit?
28. Find the largest 3-digit number which is exactly divisible by 3.
29. Find the prime numbers between 90 and 100.
30, 52 is a 3-digit number with as a missing digit. If the number is divisible by 6, find the least value of the missing digit.
31. What least value should be given to so that the number 63421 is divisible
by 37 32. What least value should be given to so that the number 915 26 is divisible by 92
33. Find the sum of prime numbers between 60 and 75.
34. If the HCF of two numbers is 16 and their product is 3072. Find their LCM.
35. What is the HCF of two co-prime numbers?
36. If the HCF and LCM of two numbers be 4 and 24 respectively. One of the numbers is 8, then find another number.
37. How many prime numbers are there between 1 and 30?
38. Which of the following number is not a multiple of 27:- 1,3,9 and 6?
39. Which of the given numbers 4, 2, 3 and 8 is not a factor of 12?
40. Which of the given numbers 21, 12, 17 and 39 is a prime number?
41. Which of the following pairs of numbers are co-prime?
(30, 415), (17, 68), (16, 81) and (15,100).
42. Which of the given pair of numbers (7, 15), (12, 49), (18, 23) and (12, 21) are not co-prime?
43. What is the HCF of 75, 60 and 210?
44. The numbers which have more than two factors are called ............
45. If A is the 5th prime number and B is the 7th prime number then how what is B-A?
46. Four bells ring at intervals of 6, 7, 8 and 9 seconds respectively. After many seconds do all the bells ring together?
47. What is the greatest number that divides 37, 50 and 123 leaving remainders 1, 2 and 3 respectively?
48. What is the least value that should be given to? so that the number 653? 47 is divisible by 11?
49. The LCM of 64 and 48 is 192. What is the HCF of these numbers?
50. Find the sum of 1+3+5+7+9+11+13+15?
Answers:
1.
3
26.
9
2.
15
27.
0
3.
(3,5), (5,7), (11,13)
28.
999
4.
None
29.
97
9996
30.
2
6.
97
311
2
7.
12, 24, 36, 48, 60
32.
4
8.
1
33.
272
9.
60
34.
192
10.
70
35.
1
11.
6, 12, 18, 24, 30, 36, 42, 48, 54
36.
12
12.
2, 3, 5, 7, 11, 13
37.
10
13.
1,3,9, 27
38.
6
14.
5,2
39.
8
15.
X-2, Y-30, 7-5
40.
17
16.
151
41.
(16, 81)
17.
12240cm
42.
(12, 21)
18.
17
43.
15
19.
7500
44.
Composite numbers
20.
75cm
45.
6
21.
9, 18, 27, 36, 45
46.
504 seconds
22.
12
47.
12
23.
1,2,4
48.
1
24.
1008
49.
16
25.
180
50.
64
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