Download Remedial - MoreMaths

1. Revision
Description
Recall basics of
factors and multiples.
Reflect and Review
1 is neither prime nor
composite, since it has only
one factor, i.e., itself.
2. Prime Factorisation
Description

Writing a number as
a product of its prime
factors is called
Prime Factorisation.
Reflect and Review

Factorisation of 36
is as follows
36
2×⑱



Prime factorisation of
a number is unique.
Prime factorisation of
a number can be
done by dividing it
with prime numbers
by which it is
divisible, till we get
quotient as a prime
number.
The number 1 should
not be included in the
prime factorisation of
any number.
2×⑨
3×3

Prime factorisation
of 720 is as follows:
2
2
2
2
3
3
5
720
360
180
90
45
15
5
1
Thus, 720 = 2 × 2 × 2
× 2 × 3 × 3 × 5.
Teasers
Answers
1) Find the smallest
number which is odd
and also prime.
2) Find the number
which is even and also
prime.
3) Find the smallest
number which is odd
and also composite.
Teasers
1) Draw the
factor tree
of 28.
2) Write the
prime
factorisation
of 2304.
1) 3
2) 2
3) 9
Answers
1)
28
2×⑭
2 × 7
2)
2
2
2
2
2
2
2
2
3
3
2304
1152
576
288
144
72
36
18
9
3
1
Thus, 2304 = 2 × 2 ×
2×2×2×2×2×2
× 3 × 3.
1
3. Common Factors and Divisibility Rules
Description
Reflect and
Review

If a number is divisible by
another number, then it is
divisible by each of the factors
of that number.

If a number is divisible by two
co-prime numbers, then it is
divisible by their product also.

If two given numbers are
divisible by a number, then
their sum is also divisible by
that number.

If two given numbers are
divisible by a number, then
their difference is also divisible
by that number.
54 is divisible by
6.
So, it is also
divisible by the
factors of 6, i.e.,
1, 2 and 3.
Teasers
Answers
1) A number is
divisible by 72.
By which other
numbers is it
also divisible?
1) 1, 2, 3, 4, 6,
8, 9, 12, 18,
24 and 36.
2) A number is
divisible by 5
and 7. By
which other
number is it
also divisible?
4. Highest Common Factor (HCF)
Description
Reflect and Review
HCF of a set of
numbers is the
product of the
common prime
factors of those
numbers.
To find the HCF of 50, 75 and 25:
2 50
5 25
5 5
1
3 75
5 25
5 5
1
5 25
5 5
1
So,
50 = 2 × 5 × 5
75 = 3 × 5 × 5
25 = 5 × 5
Thus, HCF of 50, 75 and 25 is 5 × 5 =
25.
2
2) 35.
Teasers
1) Find the HCF of
the following:
a) 162, 216,
270,
b) 135, 270,
360.
2) Find the greatest
number that will
divide 125, 157 and
220 leaving
remainders 1, 2 and
3 respectively.
Answers
1) a) 54
b) 45
2) 31
For easy solving in
case of bigger
numbers, the HCF
can be calculated by
dividing the bigger
numbers by the
common prime
factors.
To find the HCF of 64, 96 and 144:
2 64, 96, 144
2 32, 48, 72
2 16, 24, 36
2 8, 12, 18
4, 6, 9
Thus, HCF of 64, 96 and 144
= 2 × 2 × 2 × 2 = 16.
5. Least Common Multiple (LCM)
Description
Reflect and Review
By Prime
factorisation, LCM is
the product of
common and noncommon prime
factors. Only one
factor should be taken
in the place of factors
common to at least
two of the given
numbers.
To find the LCM of 12, 8 and 15:
By Common division
method, LCM is
calculated by finding
the product of all the
divisors till the
numbers in the rows
are all 1.
To find the LCM of 48, 72 and
108:
12 = 2 × 2 × 3
8=2×2×2
15 = 3 × 5
Thus, LCM of 12, 8 and 15
= 2 × 2 × 2 × 3 × 5 = 120.
Teasers
1) Find the LCM of
the following
a) 36, 54, 72
b) 24, 30, 42
Answers
1) a) 216
b) 840
2) 10032
2) Find the smallest
5-digit number
which is divisible
by the numbers
8, 12 and 16.
2 48, 72, 108
2 24, 36, 54
2 12, 18, 27
2 6, 9, 27
3 3, 9, 27
3 1, 3, 9
3 1, 1, 3
1, 1, 1
Thus, LCM of 48, 72 and 108
= 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432.
3
6. Relation between HCF and LCM of Two Numbers
Description
Reflect and
Teasers
Review
For any two numbers,
HCF × LCM
= Product of the numbers
For two numbers, 80
and 100:
HCF = 20
LCM = 400
HCF × LCM
= 20 × 400 = 8000.
Product of the
numbers
= 80 × 100 = 8000.
Thus, HCF × LCM =
Product of the
numbers.
7. Real Life Applications
Description
Reflect and Review
The concept of
HCF and LCM can
be applied in
various real life
situations.
4

The longest possible
square tile that can be
fixed on all the sides of a
tank of dimensions 200
m, 120 m and 280 m is
40 m since, HCF of 200,
120 and 280 is 40.

If the traffic light at
three signals on the way
to Atul’s office flashes
red once in every 24 sec,
18 sec, and 32 sec
respectively, then once
in every 288 sec (i.e., in
every 4 min 48 sec), all
the three signals shows
the red light together,
since LCM of 24, 18 and
32 is 288.
Answers
1) The HCF and LCM of
1) 144
two numbers are 18
and 864 respectively. 2) 1;
If one of the numbers
Co-primes.
is 108, find the other
number.
2) If the LCM of two
numbers is 1860 and
their product is also
1860, then find their
HCF. Can you guess
the type for the two
numbers?
Teasers
Answers
1) Gina is making flower
1) 8
arrangements. She has
16 roses and 72 daisies. 2) 360 cm ×
If Gina wants to make
60 cm
all the arrangements
identical and have no
flowers left over, what
is the greatest number
of flower arrangements
she can make?
2) Rectangular tiles of
dimensions 24 cm × 15
cm, 36 cm × 10 cm and
30 × 20 cm are
available in a shop.
What is the minimum
length of a square hall
in which you can place
all the three types of
tiles separately without
breaking them?