Class 7 Mathematics

Chapter 11 Solutions — Finding Common Ground

Open Solutions PDF →Also seeTextbook page→

Open Solutions PDFReads in your browser→Also seeTextbook page→

Get the full experience in the app

Free practice, offline PDFs & solutions.

Already installed? Open in the app

Solutions

Overview

Step-by-step NCERT solutions for Finding Common Ground (Chapter 11, NCERT Class 7 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Finding Common Ground textbook chapter.

Solved

What these solutions cover

All 28 questions in Finding Common Ground are solved in the PDF. Here's what's inside, exercise by exercise:

List All Factors Using Prime Factorisation

List all the factors of 90.
List all the factors of 105.
List all the factors of 132.
List all the factors of 360. (This number has 24 factors.)
List all the factors of 840. (This number has 32 factors.)

Common Factors and HCF Using Prime Factorisation

Find the common factors and the HCF of 50 and 60.
Find the common factors and the HCF of 140 and 275.
Find the common factors and the HCF of 77 and 725.
Find the common factors and the HCF of 370 and 592.
Find the common factors and the HCF of 81 and 243.

Find HCF Using Prime Factorisation

1.
Find the HCF of the following numbers:
- (a) 24 and 180
- (b) 42, 75, and 24
- (c) 240 and 378
- (d) 400 and 2500
- (e) 300 and 800
Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Idli-Vada Game — First Common Multiple / LCM

1.
In the Idli-Vada game, find the first number at which 'idli-vada' is called for each pair:
- (a) 4 and 6
- (b) 7 and 11
- (c) 14 and 30
- (d) 15 and 55

Find LCM Using Prime Factorisation

1.
Find the LCM of the following numbers:
- (a) 30 and 72
- (b) 36 and 54
- (c) 105, 195, and 65
- (d) 222 and 370

Patterns and Properties of HCF and LCM

1.
The LCM of 3 and 24 is 24 (one of the two given numbers).
- (a) Find more such number pairs where the LCM is one of the two numbers.
- (b) Make a general statement about such numbers. Describe them using algebra.
2.
Find the HCF of the following pairs of multiples of 14:
- (a) 18×10 and 18×15
- (b) 10×38 and 10×21
- (c) 5×13 and 5×20
- (d) 12×16 and 12×20
In which of the cases above is the HCF the same as the common multiplier (like case (b) where HCF = 10)?

HCF × LCM Property and Applications

In the two rows of coloured stars, colours repeat as shown. When will the blue stars meet next? Row 1 has a repeating pattern of 6 colours; blue is at every 6th position. Row 2 has a repeating pattern of 4 colours; blue is at every 4th position.
Is 5 × 7 × 11 × 11 a multiple of 5 × 7 × 7 × 11 × 2? (b) Is 5 × 7 × 11 × 11 a factor of 5 × 7 × 7 × 11 × 2?
3.
Find the HCF and LCM of the following (state your answers in the form of prime factorisations):
- (a) 3 × 3 × 5 × 7 × 7 and 12 × 7 × 11
- (b) 45 and 36
Find two numbers whose HCF is 1 and LCM is 66.
A cowherd took all his cows to graze. They came to a crossing with 3 gates, then one with 5 gates, then one with 7 gates. At each crossing, an equal number of cows passed through each gate. If the cowherd had fewer than 200 cows, how many cows did he have?
6.
The length, width, and height of a box are 12 cm, 18 cm, and 36 cm respectively. Which of the following sized cubes can be packed in this box without leaving gaps?
- (a) 9 cm
- (b) 6 cm
- (c) 4 cm
- (d) 3 cm
- (e) 2 cm
7.
Among the numbers below, which is the largest number that perfectly divides both 306 and 36?
- (a) 36
- (b) 612
- (c) 18
- (d) 3
- (e) 2
- (f) 360
Find the smallest number that is divisible by 3, 4, 5, and 7, but leaves a remainder of 10 when divided by 11.
9.
Children are playing 'Fire in the Mountain'. When 6 was called out, no one got out. When 9 was called out, no one got out. But when 10 was called out, some people got out. How many children could have been playing initially?
- (a) 72
- (b) 90
- (c) 45
- (d) 3
- (e) 36
- (f) None of these
10.
Tick the correct statement(s). The LCM of two different prime numbers (m, n) can be:
- (a) Less than both numbers
- (b) In between the two numbers
- (c) Greater than both numbers
- (d) Less than m × n
- (e) Greater than m × n
What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, and 10?

Sample

Worked examples

3 questions from Finding Common Ground, solved step by step.

List All Factors Using Prime Factorisation

List all the factors of 90.

Prime factorise 90:

90 = 2 × 3 × 3 × 5

Every factor is a 'subpart' of this factorisation. List systematically:

Single prime factors: 2, 3, 5

Pairs: 2 × 3 = 6, 3 × 3 = 9, 2 × 5 = 10, 3 × 5 = 15

Triples: 2 × 3 × 3 = 18, 2 × 3 × 5 = 30, 3 × 3 × 5 = 45

All four: 2 × 3 × 3 × 5 = 90

Include 1.

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Answer: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Common Factors and HCF Using Prime Factorisation

Find the common factors and the HCF of 50 and 60.

Prime factorisations:

50 = 2 × 5 × 5

60 = 2 × 2 × 3 × 5

Common subparts (prime factors appearing in both):

2 appears at least once in both → 2 is a common factor.

5 appears at least once in both → 5 is a common factor.

2 × 5 = 10 is a common subpart → 10 is a common factor.

Common factors: 1, 2, 5, 10

HCF = the largest common subpart = 2 × 5 = 10

Answer: Common factors: 1, 2, 5, 10. HCF = 10

Find HCF Using Prime Factorisation

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

No, we cannot say that. Factorising into composite numbers (like 6 × 12) does not give the prime factorisation.

The correct prime factorisations are:

72 = 2 × 2 × 2 × 3 × 3

144 = 2 × 2 × 2 × 2 × 3 × 3

Common primes: 2 (min 3 times), 3 (min 2 times).

HCF = 2 × 2 × 2 × 3 × 3 = 72

So 72 has many common factors with 144 (in fact, 72 itself divides 144).

Composite factorisation hides common prime factors inside the composite factors.

Answer: No. Prime factorisations show 72 = 2³ × 3² and 144 = 2⁴ × 3², so HCF = 72. The composite factorisation hides common factors inside 6, 12, 8, and 18.

All 28 questions in Finding Common Ground are solved step by step in the Solutions PDF above.

Keep solving

Chapter 11 Solutions — Finding Common Ground

Overview

What these solutions cover

List All Factors Using Prime Factorisation

Common Factors and HCF Using Prime Factorisation

Find HCF Using Prime Factorisation

Idli-Vada Game — First Common Multiple / LCM

Find LCM Using Prime Factorisation

Patterns and Properties of HCF and LCM

HCF × LCM Property and Applications

Worked examples

List all the factors of 90.

Find the common factors and the HCF of 50 and 60.

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

More solutions in Ganita Prakash

More NCERT Solutions for Class 7