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# Highest Common factor

↪ **Common Factors** – Factors in common between two or more numbers are called common factors of those numbers. e.g.,

Factors of 4 are __1__, __2__ and __4__

Factors of 12 are __1__, __2__, 3, __4__, 6 and 12

Factors of 16 are __1__, __2__, __4__, 8 and 16

∴ Common factors of 4, 12 and 16 are 1, 2 and 4.

↪ Among these common factors, 4 is the highest or greatest (4 > 2 > 1), so we can say that the highest common factor of 4, 12 and 16 is 4.

*“The Highest Common factor (HCF) or Greatest Common Divisor of two or more numbers is the highest (or greatest ) of their common factors. “*

## Prime factorisation method to find HCF

↪ First the numbers are factorised into their prime factors.

↪ Then the prime factors in common with their least occurrence is taken and multiplied to get the required HCF.

e.g.,

HCF of 12, 45 and 75 is found as follows –

12= 2×2×**3**,

45= **3**×5×5,

75= **3**×5×5,

The common factor of 12, 45, and 75 is 3 (occuring only once). Thus, the HCF of 12, 45 and 75 is 3.

# Lowest Common multiple

**Common Multiples**: Multiples in common between two or more numbers are called common multiples. e.g.,

Multiples of 4 are – 4, 8, 12, 16, 20, 24, **28**, 32, 36, 40, 44, 48, 52, **56**, 60, …

Multiples of 7 are – 7, 14, 21, **28**, 35, 42, 49, **56**, 63, …

Common multiples of 4 and 7 are 28, 56, …

Among the common multiples, 28 is the lowest (28 < 56 <…), so we can say that the Lowest common multiple of 4 and 7 is 28..

*“The Lowest Common Multiple (LCM) of two or more numbers is the lowest (or smallest or least) of their common multiples. “*

## Prime factorisation method to find LCM

↪ First the numbers are factorised into their prime factors.

↪ Then the each prime factors with their maximum occurrence is taken and multiplied to get the required LCM*. *

e.g.,

LCM of 40, 48 and 45 is found as follows –

40 = 2×2×2×**5**

48 = **2**×**2**×**2**×**2**×3

45 = **3**×**3**×**5**

Maximum occurrence of prime factors 2, 3 and 5 is four, two and one respectively.

∴ Required LCM = __2____×2×2×2__×__3×3__×__5__** =** 720

#### Time to think

1) What is the HCF of two consecutive-

(a) numbers (b) even numbers (c) odd numbers.

2) HCF of two co-prime numbers (4 = 2×2 & 15 = 3×5) found to be 0 using prime factorisation method, since there is no common prime factor. Is the answer correct? If not, what is the correct HCF?

#### Answer

1) (a) 1 (b) 2 (c) 1

2) No, correct answer is 1 which is only common factor co-primes have but it is not a prime number that’s why we don’t find it in prime factorisation method.

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