Topic - Numbers-1

Important Concepts and Formulas 

Topic - Numbers

• Number Sets and Properties of Numbers   



• Counting Numbers (Natural numbers) : 1, 2, 3 ...


• Whole Numbers : 0, 1, 2, 3 ...


• Integers : -3, -2, -1, 0, 1, 2, 3 ...


• Rational Numbers
Rational numbers can be expressed as ab where a and b are integers and b≠0
Examples: 112, 42, 0, −811 etc.
All integers, fractions and terminating or recurring decimals are rational numbers.



• Irrational Numbers
Any number which is not a rational number is an irrational number. In other words, an irrational number is a number which cannot be expressed as ab where a and b are integers.
For instance, numbers whose decimals do not terminate and do not repeat cannot be written as a fraction and hence they are irrational numbers.
Example : π, 2√, (3+5√), 43√ (meaning 4×3√), 6√3 etc
Please note that the value of π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679...
We cannot π as a simple fraction (The fraction 22/7 = 3.14.... is just an approximate value of π)


• Real Numbers
Real numbers include counting numbers, whole numbers, integers, rational numbers and irrational numbers.


• Surds
Let a be any rational number and n be any positive integer such that a√n is irrational. Then a√n is a surd.
Example : 3√, 10−−√6, 43√ etc
Please note that numbers like 9√, 27−−√3 etc are not surds because they are not irrational numbers
Every surd is an irrational number. But every irrational number is not a surd. (eg : π , e etc are not surds though they are irrational numbers.)

• Addition, Subtraction and Multiplication Rules for Even and Odd Numbers    



Addition Rules for Even and Odd Numbers
• The sum of any number of even numbers is always even


• The sum of even number of odd numbers is always even


• The sum of odd number of odd numbers is always odd


Subtraction Rules for Even and Odd Numbers
• The difference of two even numbers is always even


• The difference of two odd numbers is always even


Multiplication Rules for Even and Odd Numbers
• The product of even numbers is always even


• The product of odd numbers is always odd


• If there is at least one even number multiplied by any number of odd numbers, the product is always even

• Divisibility    



• Divisible By
One whole number is divisible by another if the remainder we get after the division is zero.
Examples
• 36 is divisible by 4 because 36 ÷ 4 = 9 with a remainder of 0.
• 36 is divisible by 6 because 36 ÷ 6 = 6 with a remainder of 0.
• 36 is not divisible by 5 because 36 ÷ 5 = 7 with a remainder of 1.
• Divisibility Rules
By using divisibility rules we can easily find out whether a given number is divisible by another number without actually performing the division. This helps to save time especially when working with numbers.


Divisibility Rule	Description	Examples
Divisibility by 2	A number is divisible by 2 if the last digit is even. i.e., if the last digit is 0 or 2 or 4 or 6 or 8	Example1: Check if 64 is divisible by 2. The last digit of 64 is 4 (even). Hence 64 is divisible by 2 Example2: Check if 69 is divisible by 2. The last digit of 69 is 9 (not even). Hence 69 is not divisible by 2
Divisibility by 3	A number is divisible by 3 if the sum of the digits is divisible by 3 (Please note that we can apply this rule to the answer again and again if we need)	Example1: Check if 387 is divisible by 3. 3 + 8 + 7 = 18. 18 is divisible by 3. Hence 387 is divisible by 3 Example2: Check if 421 is divisible by 3. 4 + 2 + 1 = 7. 7 is not divisible by 3. Hence 421 is not divisible by 3
Divisibility by 4	A number is divisible by 4 if the number formed by the last two digits is divisible by 4.	Example1: Check if 416 is divisible by 4. Number formed by the last two digits = 16. 16 is divisible by 4. Hence 416 is divisible by 4 Example2: Check if 481 is divisible by 4. Number formed by the last two digits = 81. 81 is not divisible by 4. Hence 481 is not divisible by 4
Divisibility by 5	A number is divisible by 5 if the last digit is either 0 or 5.	Example1: Check if 305 is divisible by 5. Last digit is 5. Hence 305 is divisible by 5. Example2: Check if 420 is divisible by 5. Last digit is 0. Hence 420 is divisible by 5. Example3: Check if 312 is divisible by 5. Last digit is 2. Hence 312 is not divisible by 5.
Divisibility by 6	A number is divisible by 6 if it is divisible by both 2 and 3.	Example1: Check if 546 is divisible by 6. 546 is divisible by 2. 546 is also divisible by 3. (Check the divisibility rule of 2 and 3 to find out this) Hence 546 is divisible by 6 Example2: Check if 633 is divisible by 6. 633 is not divisible by 2 though 633 is divisible by 3. (Check thedivisibility rule of 2 and 3 to find out this) Hence 633 is not divisible by 6 Example3: Check if 635 is divisible by 6. 635 is not divisible by 2. 635 is also not divisible by 3. (Check thedivisibility rule of 2 and 3 to find out this) Hence 635 is not divisible by 6 Example4: Check if 428 is divisible by 6. 428 is divisible by 2 but 428 is not divisible by 3.(Check the divisibility rule of 2 and 3 to find out this) Hence 428 is not divisible by 6
Divisibility by 7	To find out if a number is divisible by 7, double the last digit and subtact it from the number formed by the remaining digits. Repeat this process until we get at a smaller number whose divisibility we know. If this smaller number is 0 or divisible by 7, the original number is also divisible by 7.	Example1: Check if 349 is divisible by 7. Given number = 349 34 - (9 × 2) = 34 - 18 = 16 16 is not divisible by 7. Hence 349 is not divisible by 7 Example2: Check if 364 is divisible by 7. Given number = 364 36 - (4 × 2) = 36 - 8 = 28 28 is divisible by 7. Hence 364 is also divisible by 7 Example3: Check if 3374 is divisible by 7. Given number = 3374 337 - (4 × 2) = 337 - 8 = 329 32 - (9 × 2) = 32 - 18 = 14 14 is divisible by 7. Hence 329 is also divisible by 7. Hence 3374 is also divisible by 7.
Divisibility by 8	A number is divisible by 8 if the number formed by the last three digits is divisible by 8.	Example1: Check if 7624 is divisible by 8. The number formed by the last three digits of 7624 = 624. 624 is divisible by 8. Hence 7624 is also divisible by 8. Example2: Check if 129437464 is divisible by 8. The number formed by the last three digits of 129437464 = 464. 464 is divisible by 8. Hence 129437464 is also divisible by 8. Example3: Check if 737460 is divisible by 8. The number formed by the last three digits of 737460 = 460. 460 is not divisible by 8. Hence 737460 is also not divisible by 8.
Divisibility by 9	A number is divisible by 9 if the sum of its digits is divisible by 9. (Please note that we can apply this rule to the answer again and again if we need)	Example1: Check if 367821 is divisible by 9. 3 + 6 + 7 + 8 + 2 + 1 = 27 27 is divisible by 9. Hence 367821 is also divisible by 9. Example2: Check if 47128 is divisible by 9. 4 + 7 + 1 + 2 + 8 = 22 22 is not divisible by 9. Hence 47128 is not divisible by 9. Example3: Check if 4975291989 is divisible by 9. 4 + 9+ 7 + 5 + 2 + 9 + 1 + 9 + 8 + 9= 63 Since 63 is big, we can use the same method to see if it is divisible by 9. 6 + 3 = 9 9 is divisible by 9. Hence 63 is also divisible by 9. Hence 4975291989 is also divisible by 9.
Divisibility by 10	A number is divisible by 10 if the last digit is 0.	Example1: Check if 2570 is divisible by 10. Last digit is 0. Hence 2570 is divisible by 10. Example2: Check if 5462 is divisible by 10. Last digit is not 0. Hence 5462 is not divisible by 10
Divisibility by 11	To find out if a number is divisible by 11, find the sum of the odd numbered digits and the sum of the even numbered digits. Now substract the lower number obtained from the bigger number obtained. If the number we get is 0 or divisible by 11, the original number is also divisible by 11.	Example1: Check if 85136 is divisible by 11. 8 + 1 + 6 = 15 5 + 3 = 8 15 - 8 = 7 7 is not divisible by 11. Hence 85136 is not divisible by 11. Example2: Check if 2737152 is divisible by 11. 2 + 3 + 1 + 2 = 8 7 + 7 + 5 = 19 19 - 8 = 11 11 is divisible by 11. Hence 2737152 is also divisible by 11. Example3: Check if 957 is divisible by 11. 9 + 7 = 16 5 = 5 16 - 5 = 11 11 is divisible by 11. Hence 957 is also divisible by 11. Example4: Check if 9548 is divisible by 11. 9 + 4 = 13 5 + 8 = 13 13 - 13 = 0 We got the difference as 0. Hence 9548 is divisible by 11.
Divisibility by 12	A number is divisible by 12 if the number is divisible by both 3 and 4	Example1: Check if 720 is divisible by 12. 720 is divisible by 3 and 720 is also divisible by 4. (Check the divisibility rules of 3 and 4 to find out this) Hence 720 is also divisible by 12 Example2: Check if 916 is divisible by 12. 916 is not divisible by 3 , though 916 is divisible by 4.(Check thedivisibility rules of 3 and 4 to find out this) Hence 916 is not divisible by 12 Example3: Check if 921 is divisible by 12. 921 is divisible by 3. But 921 is not divisible by 4.(Check the divisibility rules of 3 and 4 to find out this) Hence 921 is not divisible by 12 Example4: Check if 827 is divisible by 12. 827 is not divisible by 3. 827 is also not divisible by 4.(Check thedivisibility rules of 3 and 4 to find out this) Hence 827 is not divisible by 12
Divisibility by 13	To find out if a number is divisible by 13, multiply the last digit by 4 and add it to the number formed by the remaining digits. Repeat this process until we get at a smaller number whose divisibility we know. If this smaller number is divisible by 13, the original number is also divisible by 13.	Example1: Check if 349 is divisible by 13. Given number = 349 34 + (9 × 4) = 34 + 36 = 70 70 is not divisible by 13. Hence 349 is not divisible by 349 Example2: Check if 572 is divisible by 13. Given number = 572 57 + (2 × 4) = 57 + 8 = 65 65 is divisible by 13. Hence 572 is also divisible by 13 Example3: Check if 68172 is divisible by 13. Given number = 68172 6817 + (2 × 4) = 6817 + 8 = 6825 682 + (5 × 4) = 682 + 20 = 702 70 + (2 × 4) = 70 + 8 = 78 78 is divisible by 13. Hence 68172 is also divisible by 13. Example4: Check if 651 is divisible by 13. Given number = 651 65 + (1 × 4) = 65 + 4 = 69 69 is not divisible by 13. Hence 651 is not divisible by 13
Divisibility by 14	A number is divisible by 14 if it is divisible by both 2 and 7.	Example1: Check if 238 is divisible by 14 238 is divisible by 2 . 238 is also divisible by 7. (Please check thedivisibility rule of 2 and 7 to find out this) Hence 238 is also divisible by 14 Example2: Check if 336 is divisible by 14 336 is divisible by 2 . 336 is also divisible by 7. (Please check thedivisibility rule of 2 and 7 to find out this) Hence 336 is also divisible by 14 Example3: Check if 342 is divisible by 14. 342 is divisible by 2 , but 342 is not divisible by 7.(Please check thedivisibility rule of 2 and 7 to find out this) Hence 342 is not divisible by 12 Example4: Check if 175 is divisible by 14. 175 is not divisible by 2 , though it is divisible by 7.(Please check thedivisibility rule of 2 and 7 to find out this) Hence 175 is not divisible by 14 Example5: Check if 337 is divisible by 14. 337 is not divisible by 2 and also by 7 (Please check the divisibility rule of 2 and 7 to find out this) Hence 337 is not divisible by 14
Divisibility by 15	A number is divisible by 15 If it is divisible by both 3 and 5.	Example1: Check if 435 is divisible by 15 435 is divisible by 3 . 435 is also divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this) Hence 435 is also divisible by 15 Example2: Check if 555 is divisible by 15 555 is divisible by 3 . 555 is also divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this) Hence 555 is also divisible by 15 Example3: Check if 483 is divisible by 15. 483 is divisible by 3 , but 483 is not divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this) Hence 483 is not divisible by 15 Example4: Check if 485 is divisible by 15. 485 is not divisible by 3 , though it is divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this) Hence 485 is not divisible by 15 Example5: Check if 487 is divisible by 15. 487 is not divisible by 3 . It is also not divisible by 5 (Please check thedivisibility rule of 3 and 5 to find out this) Hence 487 is not divisible by 15
Divisibility by 16	A number is divisible by 16 if the number formed by the last four digits is divisible by 16.	Example1: Check if 5696512 is divisible by 16. The number formed by the last four digits of 5696512 = 6512 6512 is divisible by 16. Hence 5696512 is also divisible by 16. Example2: Check if 3326976 is divisible by 16. The number formed by the last four digits of 3326976 = 6976 6976 is divisible by 16. Hence 3326976 is also divisible by 16. Example3: Check if 732374360 is divisible by 16. The number formed by the last three digits of 732374360 = 4360 4360 is not divisible by 16. Hence 732374360 is also not divisible by 16.
Divisibility by 17	To find out if a number is divisible by 17, multiply the last digit by 5 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 17, the original number is also divisible by 17.	Example1: Check if 500327 is divisible by 17. Given Number = 500327 50032 - (7 × 5 )= 50032 - 35 = 49997 4999 - (7 × 5 ) = 4999 - 35 = 4964 496 - (4 × 5 ) = 496 - 20 = 476 47 - (6 × 5 ) = 47 - 30 = 17 17 is divisible by 17. Hence 500327 is also divisible by 17 Example2: Check if 521461 is divisible by 17. Given Number = 521461 52146 - (1 × 5 )= 52146 -5 = 52141 5214 - (1 × 5 ) = 5214 - 5 = 5209 520 - (9 × 5 ) = 520 - 45 = 475 47 - (5 × 5 ) = 47 - 25 = 22 22 is not divisible by 17. Hence 521461 is not divisible by 17
Divisibility by 18	A number is divisible by 18 if it is divisible by both 2 and 9.	Example1: Check if 31104 is divisible by 18. 31104 is divisible by 2. 31104 is also divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this) Hence 31104 is divisible by 18 Example2: Check if 1170 is divisible by 18. 1170 is divisible by 2. 1170 is also divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this) Hence 1170 is divisible by 18 Example3: Check if 1182 is divisible by 18. 1182 is divisible by 2 , but 1182 is not divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this) Hence 1182 is not divisible by 18 Example4: Check if 1287 is divisible by 18. 1287 is not divisible by 2 though it is divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this) Hence 1287 is not divisible by 18
Divisibility by 19	To find out if a number is divisible by 19, multiply the last digit by 2 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 19, the original number is also divisible by 19.	Example1: Check if 74689 is divisible by 19. Given Number = 74689 7468 + (9 × 2 )= 7468 + 18 = 7486 748 + (6 × 2 ) = 748 + 12 = 760 76 + (0 × 2 ) = 76 + 0 = 76 76 is divisible by 19. Hence 74689 is also divisible by 19 Example2: Check if 71234 is divisible by 19. Given Number = 71234 7123 + (4 × 2 )= 7123 + 8 = 7131 713 + (1 × 2 )= 713 + 2 = 715 71 + (5 × 2 )= 71 + 10 = 81 81 is not divisible by 19. Hence 71234 is not divisible by 19
Divisibility by 20	A number is divisible by 20 if it is divisible by 10 and the tens digit is even. (There is one more rule to see if a number is divisible by 20 which is given below. A number is divisible by 20 if the number is divisible by both 4 and 5)	Example1: Check if 720 is divisible by 20 720 is divisible by 10. (Please check the divisibility rule of 10 to find out this). The tens digit = 2 = even digit. Hence 720 is also divisible by 20 Example2: Check if 1340 is divisible by 20 1340 is divisible by 10. (Please check the divisibility rule of 10 to find out this). The tens digit = 2 = even digit. Hence 1340 is divisible by 20 Example3: Check if 1350 is divisible by 20 1350 is divisible by 10. (Please check the divisibility rule of 10 to find out this). But the tens digit = 5 = not an even digit. Hence 1350 is not divisible by 20 Example4: Check if 1325 is divisible by 20 1325 is not divisible by 10 (Please check the divisibility rule of 10 to find out this) though the tens digit = 2 = even digit. Hence 1325 is not divisible by 20



• What are Factors of a Number and how to find it out?   



• Factors of a number
  If one number is divisible by a second number, the second number is a factor of the first number
  The lowest factor of any positive number = 1
  The highest factor of any positive number = the number itself
  Example
  The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0
• How to find out factors of a number
  • Write down 1 and the number itself (lowest and highest factors).
  • Check if the given number is divisible by 2 (Reference: Divisibility by 2 rule)
    If the number is divisible by 2, write down 2 as the second lowest factor and divide the given number by 2 to get the second highest factor
  • Check for divisibility by 3, 4,5, and so on. till the beginning of the list reaches the end
  Example1: Find out the factors of 72
  • Write down 1 and the number itself (72) as lowest and highest factors.
  1 . . . 72
  
  
  • 72 is divisible by 2 (Reference: Divisibility by 2 Rule).
    72 ÷ 2 = 36. Hence 2^nd lowest factor = 2 and 2^nd highest factor = 36. So we can write as
  1, 2 . . . 36, 72
  
  
  • 72 is divisible by 3 (Reference: Divisibility by 3 Rule).
    72 ÷ 3 = 24 . Hence 3^rd lowest factor = 3 and 3^rd highest factor = 24. So we can write as
  1, 2, 3, . . . 24, 36, 72
  
  
  • 72 is divisible by 4 (Reference: Divisibility by 4 Rule).
    72 ÷ 4 = 18. Hence 4^th lowest factor = 4 and 4^th highest factor = 18. So we can write as
  1, 2, 3, 4, . . . 18, 24, 36, 72
  
  
  • 72 is not divisible by 5 (Reference: Divisibility by 5 Rule)
  
  
  • 72 is divisible by 6 (Reference: Divisibility by 6 Rule).
    72 ÷ 6 = 12. Hence 5^th lowest factor = 6 and 5^th highest factor = 12. So we can write as
  1, 2, 3, 4, 6, . . . 12, 18, 24, 36, 72
  
  
  • 72 is not divisible by 7 (Reference: Divisibility by 7 Rule)
  
  
  • 72 is divisible by 8 (Reference: Divisibility by 8 Rule).
    72 ÷ 8 = 9. Hence 6^th lowest factor = 8 and 6^th highest factor = 9.
  Now our list is complete and the factors of 72 are
  1, 2, 3, 4, 6, 8, 9 12, 18, 24, 36, 72
  Example2: Find out the factors of 22
  • Write down 1 and the number itself (22) as lowest and highest factors
  1 . . . 22
  • 22 is divisible by 2 (Reference: Divisibility by 2 Rule).
    22 ÷ 2 = 11. Hence 2^nd lowest factor = 2 and 2^nd highest factor = 11. So we can write as
  1, 2 . . . 11, 22
  • 22 is not divisible by 3 (Reference: Divisibility by 3 Rule).
  • 22 is not divisible by 4 (Reference: Divisibility by 4 Rule).
  • 22 is not divisible by 5 (Reference: Divisibility by 5 Rule).
  • 22 is not divisible by 6 (Reference: Divisibility by 6 Rule).
  • 22 is not divisible by 7 (Reference: Divisibility by 7 Rule).
  • 22 is not divisible by 8 (Reference: Divisibility by 8 Rule).
  • 22 is not divisible by 9 (Reference: Divisibility by 9 Rule).
  • 22 is not divisible by 10 (Reference: Divisibility by 10 Rule).
  Now our list is complete and the factors of 22 are
  1, 2, 11, 22
• Important Properties of Factors
If a number is divisible by another number, then it is also divisible by all the factors of that number.

Example : 108 is divisible by 36 because 106 ÷ 38 = 3 with remainder of 0.

The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0.

Hence, 108 is also divisible by each of the numbers 1, 2, 3, 4, 6, 9, 12, 18, 36.

• What are Prime Numbers and Composite Numbers?   



• Prime Numbers
A prime number is a positve integer that is divisible by itself and 1 only. Prime numbers will have exactly two integer factors.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.
Please note the following facts
Zero is not a prime number because zero is divisible by more than two factors. Zero can be divided by 1, 2, 3 etc.
(0 ÷ 1 = 0, 0÷ 2 = 0 ...)
One is not a prime number because it does not have two factors. It is divisible by only 1
• Composite Numbers
Composite numbers are numbers that have more than two factors. A composite number is divisible by at least one number other than 1 and itself.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc.
Please note that zero and 1 are neither prime numbers nor composite numbers.
Every whole number is either prime or composite, with two exceptions 0 and 1 which are neither prime nor composite