# Simplification Techniques and Tricks

Simplification Techniques and Tricks

Classification:-

Natural Numbers (प्राकृत संख्या):

all counting numbers ( 1,2,3,4,5….∞)

Whole Numbers (पूर्ण संख्या):
natural number + zero( 0,1,2,3,4,5…∞)

Integers:
All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞)

Even & Odd Numbers:
All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞)

Prime Numbers:
It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞)

Composite Numbers:
Natural numbers which are not prime

Co-Prime:
Two natural number a and b are said to be co-prime if their HCF is 1.

Divisibility

Divisible by 2 End with 0,2,4,6,8 are divisible by 2 254,326,3546,4718 all are divisible by 2
Divisible by 3 Sum of its digits is divisible by 3 375,4251,78123 all are divisible by 3. [549=5+4+9][5+4+9=18]18 is divisible by 3 hence 549 is divisible by 3.
Divisible by 4 Last two digit divisible by 4 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4.
Divisible by 5 Ends with 0 or 5 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5.
Divisible by 6 Divides by Both 2 & 3 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6.
Divisible by 8 Last 3 digit divide by 8 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8.
Divisible by 10 End with 0 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10.
Divisible by 11 [Sum of its digit in
odd places-Sum of its digits
in even places]= 0 or multiple of 11
Consider the number 39798847

(Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)

(23-12)
23-12=11, which is divisible by 11. So 39798847 is divisible by 11.

Division and Remainder Rules
Assume we separate 45 by 6
consequently ,speak to it as:
profit = ( divisor✘quotient ) + leftover portion
on the other hand
divisior= [(dividend)- (remainder]/remainder
could be compose it as
x = kq + r where (x = dividend,k = divisor,q = quotient,r = leftover portion)
Case:
On separating a specific number by 342, we get 47 as leftover portion. In the event that the same number is separated by 18, what will be the rest of
Number = 342k + 47
( 18 ✘19k ) + ( 18 ✘2 ) + 11
18 ✘( 19k + 2 ) +11.
Leftover portion = 11
Entirety Rules
(1+2+3+………+n) = 1/2 n(n+1)
(1²+2²+3²+………+n²) = 1/6 n (n+1) (2n+1)
(1³+2³+3³+………+n³) = 1/4 n2 (n+1)2
Number juggling Progression (A.P.)
an, a + d, a + 2d, a + 3d, ….are said to be in A.P. in which first term = an and basic distinction = d.
Give the nth term a chance to be tn and keep going term = l, then
a) nth term = a + ( n – 1 ) d
b) Sum of n terms = n/2 [2a + (n-1)d]
c) Sum of n terms = n/2 (a+l) where l is the last term