Properties of HCF and LCM: For the better understanding of the concept LCM (Lowest Common Multiple) and HCF (Highest Common Factor), we need to recollect the terms multiples and factors. Letâ€™s learn about LCM, HCF, and relation between HCF and LCM of natural numbers.
Multiples: A multiple is any number which is exactly divisible by a given number. Ex: 3, 6,9,12, etc are the multiples of 3.
Factors: A factor is a number which divides any given number without leaving a remainder. Ex: 2,3,4,6,8,12 are the factors of 24.
Lowest Common Multiple (LCM): Â The least or smallest common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 10, 15, and 20 is 60.
Highest Common Factor (HCF): The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor).For example, HCF of 4, 6 and 8 is 2.
4 = 2 Ã—Â 2
6 =3 Ã—Â 2
8 = 4 Ã—Â 2
Here, highest common factor of 4, 6 and 8 is 2.
Both HCF and LCM of given numbers can be found by using two methods; they are division method and prime factorization.
HCF and LCM Properties
Property 1: The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
LCM Ã—Â HCF = Product of the Numbers
Suppose A and B are two numbers, then.
LCM (A & B) Ã—Â HCF (A & B) = A Ã—Â B
Example 1: Prove that: LCM (9 & 12) Ã—Â HCF (9 & 12) = Product of 9 and 12
Solution: LCM and HCF of 9 and 12:
9 = 3Â Ã— 3 = 3Â²
12 = 2Â Ã— 2Â Ã— 3 = 2Â²Â Ã— 3
LCM of 9 and 12 = 2Â²Â Ã— 3Â² = 4Â Ã— 9 = 36
HCF of 9 and 12 = 3
LCM (9 & 12)Â Ã— HCF (9 & 12) = 36Â Ã— 3 = 108
Product of 9 and 12 = 9Â Ã— 12 = 108
Hence, LCM (9 & 12)Â Ã— HCF (9 & 12) = 108 = 9Â Ã— 12
Property 2: HCF of coprime numbers is 1. Therefore LCM of given coprime numbers is equal to the product of the numbers.
LCM of Coprime Numbers = Product Of The Numbers
Example 2: 8 and 9 are two coprime numbers. Using this numbers verify, LCM of Coprime Numbers = Product Of The Numbers
Solution: LCM and HCF of 8 and 9:
8 = 2Â Ã— 2Â Ã— 2 = 2Â³
9 = 3Â Ã— 3 = 3Â²
LCM of 8 and 9 = 2Â³Â Ã—Â 3Â² = 8Â Ã— 9 = 72
HCF of 8 and 9 = 1
Product of 8 and 9 = 8Â Ã— 9 = 72
Hence, LCM of coprime numbers = Product of the numbers
Property 3: H.C.F. and L.C.M. of Fractions
LCM of fractions =Â \(\frac{LCM \: of\: numerators}{HCF \: of\: denominators}\)
HCF of fractions =Â \(\frac{HCF \: of\: numerators}{LCM \: of\: denominators}\)
Example 3: Find the HCF ofÂ \(\frac{12}{25}\),Â \(\frac{9}{10}\),Â \(\frac{18}{35}\),Â \(\frac{21}{40}\)
Solution: The required HCF is = \(\frac{ HCF\: of\: 12, \: 9,\: 18, \: 21}{LCM \: of\: 25, \: 10,\: 35, \: 40}\) = \(\frac{3}{1400}\)
To solve more problems on HCF and LCM download BYJU’S – The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams.
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