Natural Numbers 1,2,3,4,5…………. Sum of natural numbers n(n+1)/2 Odd Numbers: 1,3,5,7,9……………… The difference between two consecutive numbers is 2 and the numbers are not divisible by 2. Sum of odd numbers=N2 1+3+5+7+9=5*5=25 Even Numbers: 2,4,6,8……………… The difference between two consecutive numbers is 2 and the numbers are divisible by 2. Sum of Even numbers=N(N+1) 2+4+6+8+10=5*6=30 Prime numbers: The Numbers which are divisible by one and itself are called as prime numbers
INTRODUCTION
MULTIPLICATION
1 7 1 8 ---------6(7X8=56,write down 6 and carry over 5) Step-2 1 7 1 8 ----------0 6(1x8+7x1+5=20, write down 0 and carry over 2) Step-3 1 7 1 8 --------------3 0 6(1x1+2=3, write it down) Step-1
MULTIPLICATION FOR THREE DIGITS
Step-1 3 2 1 1 3 2 ------------2(1x2=2) Step-2 3 2 1 1 3 2 ----------------7 2(2x2+3x1=7) Step-3 3 2 1 1 3 2 ----------------3 7 2(2x3+3x2+1x1=13,write down 3 and carry over 1) Step-4 3 2 1 1 3 2 ------------------2 3 7 2 (3x3+1x2+1=12,write down 2 and carry over 1 Step-5 3 2 1 1 3 2 -----------------4 2 3 7 2(1x3+1=4)
MULTIPLICATION
1 1 2 +12 9 8 -2 -------- ----(112-2) (12x2) (110x100)-(24) =10976 Therefore 112*98 = 10976
SQUARING
Shortcut for finding the square of a number Example: let us calculate the product of 96*96 Here every number must be compare with the 100. See here the given number 96 which is 4 difference with the 100. so subtract 4 from the 96 we get 92 ,then the square of the number 4 it is 16 place the 16 beside the 92 we get answer 96 - 4 -------------9 2 -------------4*4=16 9216 therefore square of the two numbers 96 * 96 =9216.
SQUARING
88 -12 -----------76 ----------12*12=144 76 + 144 -------------------7744
46
4 ---------------42 as 50 = 100/2 42/2=21 now place 4*4=16 beside 21 2116
ENDING WITH 9’S
143*999=
143-1=142 999 -142 857 append 857 to 142 so the answer is 142857
SQUARING FOR THREE DIGITS USING DUPLEX COMBINATION PROCESS EXAMPLES for 2 Duplex (D) = 22 = 4 for 34 D= 2x(3x4)=24 for 103 D = 2(1x3)+02 for 1342 D= 2(1x2)+2(3x4)=28 Ex:- 207 square is D for 2/D for 20/D for207/Dfor07/D for 7 =22/2(2x0)/2(2x7)+02/2(0x7)/72 =4 /0 / 28 /0 /49 = 4 /0+2 / 8 / 0+4 /9 = 42849
SQUARES OF NUMBERS FROM 51 TO 59
We take general representative of the numbers (from 51 to 59) as 5A Now, (5A) square = 52/ 2x5xA / A2 25/10xA/ A2 We have 10xA = A0 (Like 10x4=40) 52 is 25/A0/ A2 = (25+A)/ A2 51 square is 25+1/12= 2601 52 square is 25+2/22= 2704 59 square is 25+9/92 = 3481
Square of a number with unit digit as 5
We take a general representative of such number, say,A5
Now, (A5)2 is A2/2xAx5/52 = A2/10xA/25 = A2/A0/25 (10xA = A0) = A2+A/25 = A(A+1)/25 15 square is 1 x(1+1)/25 = 225 115 square is 11 x (11+1)/25 = 13225
MULTIPLICATION OF A NUMBER ENDING WITH 11
x11= 2 (2+3) 3 = 253 56X11 = 5 (5+6) 6 = 5 6 11 -------616 56X11 = 616
23
UNIT DIGIT ADDITION IS TEN AND TEN’S DIGIT MUST BE SAME
X 27 = here unit digits are 3 and 7 and their addition is 10 and ten’s digit is 2 Then 2 after number is 3 (2x3)(3x7)=621 38 x32= (3x4)(8x2)=1216
23
CUBING
12 CUBE IS Step-1 : we see that the ten-digit in the number is 1, so we write the cube of 1 And also as the ratio between 1 and 2 is 1:2, the next digits will be double the previous one. So, the first row is 1 2 4 8 Step -2 : in the above row our 2nd and 3rd digits (from right) are 4 and 2 respectively. So, we write down 8 and 4 below 4 and 2 respectively. Then add up the two rows 1 2 4 8 4 8 ----------------1 7 2 8 -----------------
DIVISIBILITY RULES
DIVISIBILITY BY 2 RULE:-any number the last digit of which is either even or 0, is divisible by2 DIVISIBILITY BY 3 RULE:- if the sum of a number is divisible by 3, the number is divisible by 3 Ex:123 1+2+3=6 is divisible by 3 Hence 123 is divisible by 3 DIVISIBILITY BY 4 RULE:-if the last two digits of a number is divisible by 4, the number is divisible by 4. The number having two or more zeros at the end is also divisible by 4. Ex:-526428 28 is divisible by 4, therefore the number is divisible by 4
DIVISIBILITY RULES
DIVISIBILITY BY 5 RULE:- if a number ends in 5 or 0 , the number is divisible by 5 Ex:-1345 as its last digit is 5, it is divisible by 5 DIVISBILITY BY 6 RULE:-if a number is divisible by both 2 and 3 , the number is divisible by 6 Ex:-35256 is clearly divisible by 2 sum of digits =3+5+2+5+21,which is divisible by 3 Thus the given number is divisible by 6. DIVISIBILITY BY 8 RULE:- if the last three digits of a number is divisible by 8, the number is also divisible by 8.Also if the last three digits of a number are zeros, the number is divisible by 8 Ex:-1256:an 256 is divisible by 8, the number is divisible by8
DIVISIBILITY RULES
DIVISIBILITY BY 9 RULE:- if the sum of all the digits of a number is divisible by 9, the number is also divisible by 9 Ex:123 1+2+3=6 is not divisible by 9 so the number is not divisible by 9 DIVISIBLITY BY 11 RULE: if the sum of the digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11 Ex:-3245682:s1=3+4+6+2=15 s2=2+5+8=15 As s1=s2, the number is divisible by 11 DIVISIBILITY BY 12 RULE:- Any number which is divisible by both 4 and 3, is also divisible by 12