PEA202 – Lecture #1 NUMBER SYSTEM

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

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

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

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

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

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

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, 59, 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

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

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: 343. 34-(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=16---16/8=2) If 1st digit is even, the check last two digit is divisible by 8 (2224---- 24/8) 9 Sum of its digits is divisible by 9. 10 It ends with 0.

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.

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

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

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

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

GEOMETRICAL PROGRESSION   P PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

GEOMETRICAL PROGRESSION   P PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

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

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

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

Question Q1.2 1397 x 1397 = ? A. 1951609 B. 1981709 C. 18362619 D. 2031719 E. None of these P PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

Solution Q1.2 Answer: Option A Explanation: 1397 x 1397 = (1397)2 = (1400 - 3)2 = (1400)2 + (3)2 - (2 x 1400 x 3) = 1960000 + 9 - 8400 = 1960009 - 8400 = 1951609. P PEA502 Analytical Skills-II :: Vishal Ahuja PEA504A Analytical Skills-II :: Vishal Ahuja

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

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

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

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

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

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

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

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

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