Class 7 maths
DIVISIBILITY
TEST:
Test of
Divisibility by 2 : A number is divisible by 2, if
its units digit is any of the digits 0, 2, 4, 6 and 8.
Example: Each of the
numbers 24, 36, 78, 192, 310, 214166 is divisible by 2.
Test of
Divisibility by 3 : A number is divisible by 3, if
the sum of its digits is divisible by 3.
Example: (i)
Consider the number 349524.
Sum of its digits = (3 + 4 + 9 + 5 + 2 + 4) =
27, which is divisible by 3.
(ii) Consider
the number 871423
Sum of its digits = (8 + 7 + 1 + 4 + 2 + 3) =
25, which is not divisible by 3.
Test of
Divisibility by 4 : A number is divisible by 4, if
the number formed by its last two
digits is divisible by 4.
Example: (i)
Consider the number 15632.
The number formed by its last two
digits is 32, which is divisible by 4.
(ii) Consider
the number 19374.
The number formed by its last two
digits is 74, which is not divisible by
4.
Test of
Divisibility by 5 : A number is divisible by 5, if
its units digit is either 0 or 5.
Example: The numbers 245, 16260, 27915, 411115 are all divisible by 5.
Test of
Divisibility by 6 : A number is divisible by 6, if
it is divisible by 2 as well as 3.
Example: (i) Consider the number
753216. Since its units digit is 6, so it is divisible by 2 Sum of its digits =
(7 + 5 + 3 + 2 + 1 + 6) = 24, which is divisible by 3.
(ii) Consider the number
453212. Since, its units digit is 2, so it is divisible by 2. Sum of its digits = (4 + 5 + 3 + 2 + 1 + 2) =17, which
is not divisible 3
Note : If a number is divisible by two
co-primes, then it is also divisible
by their product.
Test of Divisibility by 8 : A number is divisible by 8, if the number formed by its last three
digits is divisible by 8.
Example: (i) Consider
the number 29512.
The number formed by its last three
digits is 512, which is divisible by 8.
So, 29512 is divisible by 8.
(ii) Consider
the number 16942.
The
number formed by its last three digits is 942, which is not divisible by 8
Test of
Divisibility by 9: A number is divisible by 9, if
the sum of its digits is divisible by 9.
Example: (i) Consider
the number 517248.
Sum
of its digits = (5 + 1 + 7 + 2 + 4 + 8) = 27, which is divisible by 9.
(ii) Consider
the number 641857.
Sum
of its digits = (6 + 4 + 1 + 8 + 5 + 7) = 31, which is not divisible by 9.
Test of
Divisibility by 10 : A number is divisible by 10,
if its units digit is 0.
Example: The numbers 1110, 301020, 15670, 19250 are all divisible by 10.
Test of
Divisibility by 11 : A number is divisible by 11,
if the difference between the sum
of its digits at odd places and sum of the digits at
even places is either 0 or a number
divisible by 11.
Example: (i) Consider the number 749859.
(Sum of digits at odd places) – (Sum
of digits at even places)
= (9 + 8 + 4) – (5 + 9 + 7) = 0.
(ii) Consider
the number 8192657.
(Sum of digits at odd places) – (Sum
of digits at even places)
= (7 + 6 + 9 + 8) – (5 + 2 + 1) = 30
– 8 = 22, which is divisible by 11.
(iii) Consider
the number 5702211
(Sum of digits at even places) – (Sum
of digits at odd places)
= (1 + 2 + 7) – (1 + 2 + 0 + 5) = (10
– 8) = 2, which is not divisible
by 11.
Prime Factors: A factor of a given number is called a prime factor if this factor
is a prime number.
Example: The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. Out of these 2,
3 and 7 are prime numbers. Therefore, 2, 3 and 7 are the prime factors of 42.
Common Factors: A number
which divides each one of the given numbers exactly, is called a common factor
of each of the given numbers.
Example: 4 divide each one of 212 and 356 exactly. Therefore, 4 is a common
factor of 212 and 356.
H.C.F. (HIGHEST COMMON FACTOR) OR
G.C.D. (GREATEST COMMON DIVISOR) :
H.C.F. or G.C.D. of two or more numbers is the
greatest number that divides each one of them exactly.
Example: Consider the numbers 36
and 54.
F(36)
= Set of all factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
F(54)
= Set of all factors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}
=
{1, 2, 3, 6, 9, 18}.
The
greatest number in F(36)
F(54) is 18.
Methods
of Finding the H.C.F. of Given Numbers Prime
Factorisation Method :
Suppose we have to find the H.C.F. of two or more
numbers.
Step 1.Express each one of the given numbers as the product of prime
factors.
Step 2.The product of terms containing least powers of common prime factors
gives the H.C.F. of the given numbers.
Example: Find the H.C.F. of 540
and 1008.
Solution: Resolving each of the
given numbers into
prime
factors, we get :
540
= 22
33
5
1008
= 24
32
7
=
22
32 = (4
9) = 36.
Example: Find the H.C.F. of 324,
288 and 360.
Solution: Resolving each of the
given
Numbers into prime factors,
we
get :
324
= 22
34
288
= 25
32
360
= 23
32
5
containing
least powers of common Prime factors
=
22
32 = (4
9) = 36.
L.C.M. (LEAST COMMON MULTIPLE)
The L.C.M. of
two or more numbers is the least natural number which is a multiple of each of
the given numbers.
Example: Consider the number 12
and 18.
M(12)
= Set of multiples of 12 = {12, 24, 36, 48, 60, 72, …..}
M(18)
= Set of multiples of 18 = {18, 36, 54, 72, 90, …..}
Least
of this number is 36.
Methods of
Finding the L.C.M. of Given Numbers
Prime
Factorisation Method :
Suppose
we have to find the L.C.M. of two or more numbers.
Step 1. Express each one of the given numbers as the
product of prime factors.
Step 2. The product of all the
different prime factors each raised to highest power that
appears in the prime factorisation of any of the given numbers,
gives the L.C.M. of the given numbers.
Example: Find the L.C.M. of 72
and 84 by prime factorisation method.
Solution : Resolving each of the
given numbers into
72
= 23
32
84
= 22
3
7
L.C.M.
= Product of terms containing
the
highest powers of all prime factors
= 23
32
7 = 504.
Relation
between H.C.F. and L.C.M. of two numbers :
We have : Product of two given numbers = Product of their H.C.F. and L.C.M.
SQUARES : The square of a
number is that number raised to the power 2.
Example: (i)
Square of 5 = 52 = 5
5 = 25 ;
(ii) Square
of 6 = 62 = 6
6 = 36 ;
(iii) Square
of
;
(iv) Square
of 0.2 = (0.2)2 = 0.2
0.2 = 0.04.
Perfect
Squares : A natural number is called a perfect square,
if it is the square of some other natural number.
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