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, 15  2016
before the end of the play
English
Verbs that take either infinitive or
gerund with no change of
meaning as follows
Start, intend, bother, begin, continue,
propose, like, prefer, hate
a He started to leave the theatre
before the end of the play
a He started leaving the theatre
15
7

a He intended to help her / he
intends helping her
a I always prefer to travel to my
village by train
a I always prefer traveling to my
village by train
Remember/forget/regret are used
with gerunds for the memories
of the past. In these constructions
the action is followed by the act
of remembering, forgetting and
regretting.
a I regret spending on her (Spend
ing is the 1st and regret is the
2nd action)
a I remember attending the marr
-iage years back. (attending is
the 1st and remembering is the
2nd action)
The sum of first five multiples of 6 is
Making number as directed
G. Janardhan Reddy
Director
Reyan coaching centre
Nagole, Hyd
9030878078
1) Every digit in a particular greatest number in ‘9’
Ex : The greatest one – digit number is ‘9’
a The greatest 2 digit number is ‘99’ and so on ….
2) In a particular smallest number, the first
digit on the left is 1 and the remaining
digits are ‘0’.
a There are 10 one digit numbers (0 to 9)
a There are 90 two – digit numbers (10 to 99)
a There are 900 three – digit numbers
(100 to 999)
a There are 9000 four – digit numbers
(1000 to 9999)
a If we consider number from 1 to 100, the
digits from 0 to 9 occur the following
number of times respectively.
Digit
0
No.of Times occur 11
PRACTICE BITS
1
2
3
4
5
6
7
8
9
21
20
20
20
20
20
20
20
20
PRACTICE BITS
1) The difference between the greatest four
digit number and the smallest four- digit
number is
1) 1
2) 90
3) 9000
4) 8999
2) How many four digit number do we
have in number system.
1) 9999 2) 9990 3) 9000
4) 8999
3) While writing numbers from 1 to 100 how
many times does the digit ‘7’ occur
1) 9
2) 10
3) 20
4) 21
ANSWERS :
1. 4, 2. 3, 3.3 .
Tests of Divisibility :
Using the rules of divisibility we can find
the no. of factors of given number without
division.
I) Divisibility by 2,4, and 8
i) By ‘2’ :- If the last digit of a number is
either 2,4,6,8 or 0, the number is
divisible by ‘2’.
Ex : 146, 3580, 1058.
ii) By ‘4’ :- If the last two digits of a
number is multiple of 4 (or) 00 the
number is divisible by ‘4’.
Ex : 8124 (24 is multiple of 4)
iii) By ‘8’ :- If the last three digit’s of a
number is multiple of 8 or 000 the
number is divisible by ‘8’.
Ex : 56896 (896 is multiple of 8)
35000 (Last three digits are 000)
II) Divisibility by 3,6 and 9 :i) By ‘3’ :- If the sum of the digits of a
number is divisible by ‘3’, then the given
number is divisible by ‘3’.
Ex : 345 is divisible by ‘3’ = (3+4+5=12)
= 1+2 =3 is divisible by ‘3’is divisible by ‘3’.
Ex : 5490891 is divisible by ‘3’ because.
Sum of digits = 5+4+9+0+8+9+1 =36.
3+6 =9
9 is divisible by ‘3’.
ii) By ‘6’ :- If a number is divisible by both
2 and 3 then the number is divisible by ‘6’.
Ex : 5412 is divisible by ‘6’ because.
Last digit is ‘2’ then divisible by ‘2’.
Sum of all digits 5+4+1+2 =12 then
divisible by 3.
iii) By ‘9’ : - If the sum of all digits of a
number is divisible by ‘9’ the number
is divisible ‘9’.
Ex : 6372 is divisible by ‘9’ because 6+3+7+2
=18 is divisible by ‘9’.
Divisibility by 5 and 10 :
a If the units place digit of a number is 0 or
5, then the numbers is divisible by 5.
a If the units place digit of a number is 0, then
the number is divisible by 10.
PRACTICE BITS
1) Which of the following number is
divisible by ‘2’
1) 3145
2) 5169
3) 6780
4) 34269
2) Which of the following number is divisible by ‘6’.
1) 5342
2) 6170
3) 6270
4) 5164
3) The number which is divisible by ‘4’ is
1) 74926
2) 3150
3) 5466
4) 5128
4) The number 54612 is divisible by
1) only 2
2) 3 Only
3) only 9
4) 2,3,9
5) The number which is divisible both 3 and 9?
1) 74211
2) 89261
3) 89262
4) 1443
ANSWERS : 1.3, 2. 3, 3.4, 4. 4, 5.3.
FACTORS
a If a number is exactly divisible by the
another number, then the second number is
factor of first numbers.
Eg:16
16 is divisible with 1,2,4,8,16 then 1,2,4,8,16
are factors for 16
To find the factors (using of test of divisibility)
Eg: 100
1 x 100
Write up to down
2 x 50
and down to up
4 x 25
1,2,4,5,10,20,25,50,100
5 x 20
10 x 10
a 1 is the factor of every number
a Every number is a factor of itself
a Number of factors of a number are finite
(countable)
Common Factors: Factors that two numbers
have in common are called the common
factors of those numbers.
Eg: Common factors of 15 and 20
Factors of 15 = 1,3,5,15
Factors of 20 = 1,2,4,5,20
Common factors of 15 and 20 is 5
Multiples : Multiples of number is the number
obtained by multiplying it with other number.
Eg: 8 x 1 = 8, 8 x 2 = 16, 8 x 3 = 24
8, 16, 24 are multiples of 8
a Every number is a multiples a multiples of itself.
a Number of multiples of a number are infinite
(un countable)
Common Multiples : A common multiples is
a number that is a multiples of a two or
more numbers.
Eg: common multiples of 4 and 18 is 36
because
4 x 9 = 36, 18 x 2 = 36
Prime Factorization
To get prime factor a number, begin
dividing by the smallest possible prime and
continue until the quotient is a prime
number.
Eg: 1200 = 2 x 2 x 2 x 2 x 3 x 5 x 5
2 1200
2 600
2 300
2 150
3 75
5 25
5
1. Factors of 60 are ?
1) 2,3,5
2) 1,2,3,5,12,60
3) 1,2,3,4,5,6,10,12,15,20,30,60
4) 1,2,3,4,8,10,12,60
2. The number of prime factors of 105 is
1) 2
2) 3
3) 4
4) 5
3. The total numbers of factors of 81 is
1) 6
2) 5
3) 4
4) 7
4. The sum of first five multiples of 6 is
1) 70
2) 84
3) 90
4) 96
5. Prime factorization of 144
1) 4 x 3 x 12
2) 2 x 2 x 2 x 2 x16
3) 2 x 2 x 2 x 2 x 3 x 3
4) 2 x 2 x 2 x 2 x 9
6. The number of multiples of 3 which are
less than 22
1) 6
2) 7
3) 8
4) 12
7. The common factor of 20 and 24
1) 2
2) 3
3) 4
4) 6
8. Sum of first five odd multiples of 3 is
1) 45
2) 75
3) 90
4) 60
9. Prime factorization of greatest 3 digit
number is
1) 2 x 2 x 5 x 5
2) 3 x 3 x 111
3) 3 x 3 x 11
4) 3 x 3 x 3 x 37
Answers
1- 3
6 - 2
2- 2
7 - 3
3- 2
8 - 2
4- 3
9 - 4
5- 3
LCM (LEAST COMMON MULTIPLE)
LCM of two or more numbers is a number
which is common multiple of the numbers
and is the smallest among all common
multiples.
Eg: Multiples of 4 are 4,8,12,16,20,24,28,…..
Multiplex of 5 are 5,10,15,20,25,30,35,…..
Among the multiples of 4 and 5 common
multiples is 20, So 20 is LCM of 4 and 5.
Methods for Finding LCM
Prime factorization method.
Step I :Write each of the given numbers as
product of prime factors.
Step II : Find the product of the highest
powers of the prime factors, which
will be LCM
Eg: LCM of 24,72,90
24 = 2 x 2 x 2 x 3
72 = 2 x 2 x 2 x 3 x 3
90 = 2 x 3 x 3 x 5
LCM = Product of all the prime factors of
each of the given numbers which occurring
maximum numbers of times.
= 2 x 2 x 2 x 3 x 3 x 5 = 360