Finding Common Ground — Class 7 Mathematics Notes (NCERT)

Snapshot

This chapter is about the "common ground" between numbers — their shared factors and shared multiples.
A factor of a number divides it exactly; a multiple is got by multiplying it by a counting number.
The biggest shared factor is the Highest Common Factor (HCF); the smallest shared multiple is the Lowest Common Multiple (LCM).
We find HCF and LCM by listing, by prime factorisation, or by the common division method.
Key identity for two numbers: $\text{HCF}\times\text{LCM}=\text{product of the two numbers}$.
HCF answers "largest equal groups / biggest measure"; LCM answers "when do they meet / smallest common amount".
Weightage: ~6 marks/year — finding HCF and LCM, and real-life word problems using them.

Detailed Notes

1. Factors and multiples — a quick recap

A factor of a number is a number that divides it exactly, leaving no remainder. For example, the factors of $12$ are $1,2,3,4,6,12$. A multiple of a number is what you get by multiplying it by $1,2,3,\dots$ — the multiples of $4$ are $4,8,12,16,20,\dots$

$1$ is a factor of every number. Every number is a factor of itself and a multiple of itself. A number has a limited list of factors but unlimited multiples.

Worked example — list the factors of 18

Find pairs that multiply to $18$: $1\times18,\ 2\times9,\ 3\times6$. So the factors of $18$ are $1,2,3,6,9,18$.

2. Common factors and the HCF

A common factor of two or more numbers is a factor shared by all of them. The largest such factor is the Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD).

Worked example — HCF of 12 and 18 by listing

Factors of $12$: $1,2,3,4,6,12$. Factors of $18$: $1,2,3,6,9,18$. Common factors: $1,2,3,6$. The highest is $6$, so $\text{HCF}(12,18)=6$.

If the only common factor of two numbers is $1$, they are called co-prime (e.g. $8$ and $15$). Co-prime numbers have $\text{HCF}=1$.

3. Common multiples and the LCM

A common multiple of two or more numbers is a multiple shared by all of them. The smallest such multiple is the Lowest Common Multiple (LCM).

Worked example — LCM of 4 and 6 by listing

Multiples of $4$: $4,8,12,16,20,24,\dots$ Multiples of $6$: $6,12,18,24,30,\dots$ Common multiples: $12,24,\dots$ The smallest is $12$, so $\text{LCM}(4,6)=12$.

There are infinitely many common multiples but only one lowest one. The LCM is never smaller than the largest of the given numbers.

4. Prime factorisation method

Every number can be written as a product of primes (its prime factorisation). This makes finding HCF and LCM systematic.

HCF = product of the common prime factors, each taken to its smallest power.
LCM = product of all prime factors that appear, each taken to its greatest power.

Worked example — HCF and LCM of 24 and 36

$24=2^{3}\times3$ and $36=2^{2}\times3^{2}$.

Common primes are $2$ and $3$. Smallest powers: $2^{2}$ and $3^{1}$. So $\text{HCF}=2^{2}\times3=12$.

All primes at greatest powers: $2^{3}\times3^{2}=8\times9=72$. So $\text{LCM}=72$.

Worked example — HCF and LCM of 60, 72, 90

$60=2^{2}\times3\times5,\ 72=2^{3}\times3^{2},\ 90=2\times3^{2}\times5.$

Common to all three: only $2$ and $3$, smallest powers $2^{1}$ and $3^{1}$, so $\text{HCF}=2\times3=6$.

All primes at greatest powers: $2^{3}\times3^{2}\times5=8\times9\times5=360$, so $\text{LCM}=360$.

5. Common division method

This is a quick way to find the LCM (and HCF) of several numbers together. Divide all the numbers by a common prime, write the quotients below, and keep going until no two numbers share a factor.

Worked example — LCM of 12, 16, 24 by common division

Divide by $2$: $6,8,12$. Again by $2$: $3,4,6$. Again by $2$: $3,2,3$. Now divide by $3$: $1,2,1$. Then by $2$: $1,1,1$.

Multiply all the divisors: $2\times2\times2\times3\times2=48$. So $\text{LCM}(12,16,24)=48$.

For the HCF by division, divide only while all numbers are divisible by the same prime; the product of those common divisors is the HCF.

6. The HCF × LCM relationship

For any two numbers, the HCF and LCM are tied together by a neat identity.

$\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$

So if you know any three of the four quantities (the two numbers, their HCF, their LCM), you can find the fourth.

Worked example — find the LCM using the identity

Two numbers are $15$ and $20$, with $\text{HCF}=5$. Then $\text{LCM}=\dfrac{15\times20}{5}=\dfrac{300}{5}=60$.

Worked example — find an unknown number

For two numbers, $\text{HCF}=6$ and $\text{LCM}=72$. One number is $24$. The other $=\dfrac{\text{HCF}\times\text{LCM}}{24}=\dfrac{6\times72}{24}=\dfrac{432}{24}=18$.

Important: this product rule works only for two numbers, not for three or more.

7. When to use HCF and when to use LCM

The trickiest part of word problems is deciding which one you need. These signals help.

Use HCF when…	Use LCM when…
Splitting things into the largest equal groups	Finding the smallest common amount or quantity
Finding the greatest length/measure that divides everything	Finding when events will happen together again
Words like maximum, greatest, largest	Words like least, minimum, smallest, together

Rough rule of thumb: HCF makes things smaller (it is a divisor of the numbers); LCM makes things bigger (it is a multiple of the numbers).

8. Word problems using HCF

Worked example — largest tile

A floor is $84$ cm by $108$ cm. Find the side of the largest square tile that fits exactly. The tile side must divide both $84$ and $108$, and be the largest such number — that is $\text{HCF}(84,108)$. $84=2^{2}\times3\times7,\ 108=2^{2}\times3^{3}$. Common: $2^{2}\times3=12$. So the largest tile is $12$ cm $\times$ $12$ cm.

Worked example — equal stacks of books

There are $48$ Maths and $60$ Science books to be stacked so that each stack has the same number and only one subject. The largest stack size $=\text{HCF}(48,60)$. $48=2^{4}\times3,\ 60=2^{2}\times3\times5$; HCF $=2^{2}\times3=12$. So each stack has $12$ books.

9. Word problems using LCM

Worked example — bells ringing together

Three bells ring at intervals of $6$, $8$ and $12$ minutes. If they ring together now, after how long will they ring together again? Answer $=\text{LCM}(6,8,12)$. $6=2\times3,\ 8=2^{3},\ 12=2^{2}\times3$; LCM $=2^{3}\times3=24$. They ring together again after $24$ minutes.

Worked example — smallest number leaving the same remainder

Find the smallest number that, when divided by $15$ and $20$, leaves a remainder of $3$ each time. First find $\text{LCM}(15,20)$. $15=3\times5,\ 20=2^{2}\times5$; LCM $=2^{2}\times3\times5=60$. Add the remainder: the required number is $60+3=63$.

10. Common mistakes to avoid

Mixing up the methods: HCF uses the smallest powers of common primes; LCM uses the greatest powers of all primes.
Using $\text{HCF}\times\text{LCM}=$ product for three or more numbers — it works only for two.
Reading "largest" but computing the LCM (or "smallest/together" but computing the HCF).
Forgetting to add the remainder back in "same remainder" LCM problems.
Thinking the LCM can be smaller than the biggest given number — it never is.
Listing factors and missing a pair — always go up in pairs like $1\times n,\ 2\times\dots$

11. Quick revision checklist

HCF = highest shared factor; LCM = lowest shared multiple.
Prime method: HCF = common primes, smallest powers; LCM = all primes, greatest powers.
$\text{HCF}\times\text{LCM}=a\times b$ (two numbers only).
Co-prime numbers have HCF $=1$.
HCF for "largest equal groups / biggest measure"; LCM for "smallest common / meeting together".
"Same remainder" problem: required number $=$ LCM $+$ remainder.

Practice MCQs

1. The HCF of $12$ and $18$ is:

$2$
$3$
$6$
$36$

Answer: (C) Common factors are $1,2,3,6$; the highest is $6$.

2. The LCM of $4$ and $6$ is:

$2$
$12$
$24$
$10$

Answer: (B) The smallest common multiple of $4$ and $6$ is $12$.

3. Two numbers are co-prime when their HCF is:

$0$
$1$
their product
the smaller number

Answer: (B) Co-prime numbers share only the factor $1$.

4. Using prime factorisation, $\text{HCF}(24,36)$ is:

$6$
$12$
$24$
$72$

Answer: (B) $24=2^{3}\times3,\ 36=2^{2}\times3^{2}$; HCF $=2^{2}\times3=12$.

5. The LCM of $24$ and $36$ is:

$12$
$36$
$72$
$144$

Answer: (C) All primes at greatest powers: $2^{3}\times3^{2}=72$.

6. For two numbers, $\text{HCF}=5$ and the numbers are $15$ and $20$. Their LCM is:

$60$
$300$
$100$
$5$

Answer: (A) $\text{LCM}=\dfrac{15\times20}{5}=60$.

7. The largest square tile that exactly covers a floor $84$ cm by $108$ cm has side:

$6$ cm
$9$ cm
$12$ cm
$24$ cm

Answer: (C) $\text{HCF}(84,108)=12$ cm.

8. Three bells ring every $6$, $8$ and $12$ minutes. They ring together again after:

$12$ min
$24$ min
$48$ min
$2$ min

Answer: (B) $\text{LCM}(6,8,12)=24$ minutes.

9. The HCF of two co-prime numbers $8$ and $15$ is:

$1$
$8$
$15$
$120$

Answer: (A) They share only the factor $1$.

10. For two numbers, $\text{HCF}=6$, $\text{LCM}=72$, one number is $24$. The other is:

$12$
$18$
$36$
$48$

Answer: (B) Other $=\dfrac{6\times72}{24}=18$.

11. The LCM of two numbers is always:

smaller than both
equal to their HCF
at least as large as the bigger number
equal to $1$

Answer: (C) Being a common multiple, the LCM is never smaller than the larger number.

12. The HCF of $60$, $72$ and $90$ is:

$2$
$3$
$6$
$12$

Answer: (C) Common primes to all three are $2$ and $3$ (smallest powers), so HCF $=6$.

13. The smallest number that leaves remainder $3$ when divided by $15$ and $20$ is:

$60$
$63$
$57$
$83$

Answer: (B) $\text{LCM}(15,20)=60$; add the remainder: $60+3=63$.

14. To split $48$ Maths and $60$ Science books into largest equal single-subject stacks, each stack has:

$6$
$12$
$24$
$4$

Answer: (B) $\text{HCF}(48,60)=12$ books per stack.

15. The identity $\text{HCF}\times\text{LCM}=a\times b$ is valid for:

any number of integers
exactly two numbers
only prime numbers
only co-prime numbers

Answer: (B) The product rule holds only for two numbers.

Important Questions

Q1. Find the HCF and LCM of $36$ and $48$ by prime factorisation. (2 marks)

Answer: $36=2^{2}\times3^{2}$ and $48=2^{4}\times3$. HCF $=2^{2}\times3=12$. LCM $=2^{4}\times3^{2}=16\times9=144$.

Q2. The HCF of two numbers is $9$ and their LCM is $90$. If one number is $18$, find the other. (2 marks)

Answer: Other number $=\dfrac{\text{HCF}\times\text{LCM}}{18}=\dfrac{9\times90}{18}=\dfrac{810}{18}=45$.

Q3. Find the greatest number that divides $245$ and $343$ exactly. (3 marks)

Answer: This is $\text{HCF}(245,343)$. $245=5\times7^{2}$ and $343=7^{3}$. The common prime is $7$ at smallest power $7^{2}$, so HCF $=49$. The greatest such number is $49$.

Q4. Two tankers contain $850$ litres and $680$ litres of milk. Find the largest container that can measure both exactly. (3 marks)

Answer: Required $=\text{HCF}(850,680)$. $850=2\times5^{2}\times17$ and $680=2^{3}\times5\times17$. Common primes at smallest powers: $2\times5\times17=170$. The largest container is $170$ litres.

Q5. Find the smallest number which when divided by $12$, $16$ and $24$ leaves a remainder of $5$ in each case. (3 marks)

Answer: First find $\text{LCM}(12,16,24)$. $12=2^{2}\times3,\ 16=2^{4},\ 24=2^{3}\times3$; LCM $=2^{4}\times3=48$. Add the remainder: $48+5=53$. The smallest such number is $53$.

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More Class 7 Mathematics chapters

Large Numbers Around Us · Arithmetic Expressions · A Peek Beyond the Point · Expressions using Letter-Numbers · Parallel and Intersecting Lines · Number Play · A Tale of Three Intersecting Lines · Working with Fractions