1. PEA202 – Lecture #1
NUMBER SYSTEM

2. DIGITS
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
NUMERAL
A group of digits, denoting a number.

3. Th H T O 2 3 6 5
PEA504A Analytical Skills-II :: Vishal Ahuja

4. TYPES OF NUMBERS NATURAL NUMBER 1 , 2 , 3 , 4 , 5 . . . WHOLE NUMBER
All natural numbers including 0.
INTEGERS
All natural numbers, 0 & negative numbers
{…, -3, -2, -1, 0, 1, 2, 3, …}
Positive Integers {1, 2, 3, …}
Negative Integers { -1, -2, -3, …}
Non-Positive & Non-Negative Integer is 0.
PEA504A Analytical Skills-II :: Vishal Ahuja

5. TYPES OF NUMBERS PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

6. EVEN & ODD NUMBERS EVEN NUMBER No’s divisible by 2. ODD NUMBER
No’s not divisible by 2.
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

7. Facts about Even & Odd No’s
Sum / Difference of two even numbers is an even number.
Sum / Difference of two odd numbers is an even number.
Sum / Difference of an even number and an odd numbers is an odd number.
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

8. TYPES OF NUMBERS – Prime No
Quick yak:
Discuss Smallest prime no is 2 and 1 is not a prime no.
2 is the only even prime no. Rest all prime no are odd numbers.
TYPES OF NUMBERS – Prime No
A Prime No is a natural no greater than 1 that has no positive divisors other than 1 and itself.
Prime no’s upto 100 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, , 61, 67, 71, 73, 79, 83, 89, 97.
A natural number greater than 1 that is not a prime number is called a Composite No.
Two no’s a & b are said to be Co-Primes, if their HCF is 1.
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

9. How to find whether a no is Prime or not
Suppose we have to find whether 191 is prime or not.
Now 14 > √191
Prime no less than 14 are 2,3,5,7,11,13.
191 is not divisible by any of these prime no.
So 191 is a prime number.
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

10. Tests of Divisibility A no is divisible by If 2
Unit’s digit is divisible by 2. 3
Sum of its digits is divisible by 3. This rule is recycled. 4
No formed by last 2 digits is 00 or is divisible by 4. 5
Unit’s digit is either 0 or 5. 6
It is divisible by both 2 & 3. 7
If the number formed by subtracting twice the last digit with the number formed by; rest of the digits is divisible by 7. Example: (3x2) = 28 is divisible by 7.
This rule is recycled. 8
If last 3 digits are 000 or if last 3 digits is divisible by 8.
In last 3-digit number :
If 1st digit is odd, Add 4 to last 2 digit and check whether it is divisible by 8.(512—12+4= /8=2)
If 1st digit is even, the check last two digit is divisible by 8
( /8) 9
Sum of its digits is divisible by 9. 10
It ends with 0.

11. Tests of Divisibility A no is divisible by If 11
The difference of sum of its digits at odd places and sum of its digits at even places, is either 0 or a no divisible by 11. 12
It is divisible by both 3 & 4. 13
Multiply last digit by 9 and subtract from rest of the numbers and if result is multiple of 13 ,then it is divisible by 13 14
It is divisible by both 2 & 7. 15
It is divisible by both 3 & 5. 16
The no formed by last four digits is divisible by 16. 24
It is divisible by both 3 & 8. 40
It is divisible by both 5 & 8. 80
It is divisible by both 5 & 16.
Quick yak:
For division with 7 & 13, use simple ways of division.
Shortcuts are very complex.

12. Dividend = (Divisor * Quotient ) + Reminder
Division Algorithm
Dividend = (Divisor * Quotient ) + Reminder P

13. Basic Formulae (a + b)(a - b) = (a2 - b2) (a + b)2 = (a2 + b2 + 2ab)
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
(a3 + b3) = (a + b)(a2 - ab + b2)
(a3 - b3) = (a - b)(a2 + ab + b2)
(a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
When a + b + c = 0, then a3 + b3 + c3 = 3abc. P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

14. ARITHMATIC PROGRESSION
An Arithmetic Progression (A.P.) is a sequence in which the difference between any two consecutive terms is constant.
Let a = first term, d = common difference
Then nth term P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

15. Sum of an A.P. P PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

16. GEOMETRICAL PROGRESSIONP
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

17. GEOMETRICAL PROGRESSIONP
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

18. HOW TO FIND THE UNIT DIGIT OF A NUMBER
Cyclicity P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

19. Question Q 1.1 Which one of the following are prime number? A. 241
D. 571 P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

20. Solution Q1.1 Clearly, 16 > √241
Prime numbers less than 16 are 2,3,5,7,11,13.
241 is not divisible by any one of them.
So, 241 is a prime number. P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

21. Question Q1.2
1397 x 1397 = ? A B C D
E. None of these P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

22. Solution Q1.2 Answer: Option A Explanation: 1397 x 1397 = (1397)2
= ( )2
= (1400)2 + (3)2 - (2 x 1400 x 3) = = = P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

23. Question Q1.3
What least number must be added to 1056, so that the sum is completely divisible by 23 ? 2
B. 3
C. 18
D. 21
E. None of these P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

24. Solution Q1.3 Answer: Option A P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

25. Question Q1.4 Find the unit digit of 295 P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

26. Solution Q1.4 The cyclicity of 2 is 4. 21 = 2 22 = 4 23 = 8 24 = 16
Divide 95 by 4. Remainder is 3.
So, the unit digit is 8. P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

27. Practice P1.1 The largest 4 digit number exactly divisible by 88 is:
E. None of these
Quick yak:
Use concept
of Q 1.3 P
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

28. Solution P1.1 Answer: Option A Explanation:
Largest 4-digit number = 9999
Reminder of 9999 / 88 = 55
Required number = ( ) = 9944. P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

29. Practice P1.2 Find the unit digit of 999 Quick yak: Use concept
PEA504A Analytical Skills-II :: Vishal Ahuja
PEA502 Analytical Skills-II :: Vishal Ahuja

30. Solution P1.2 The cyclicity of 9 is 2. 91 = 9 92 = 81
Divide 99 by 2. Remainder is 1.
So, the unit digit is 9. P
PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja

31. Next Class: HCF & LCM P PEA502 Analytical Skills-II :: Vishal Ahuja
PEA504A Analytical Skills-II :: Vishal Ahuja