1. Write properties of HCF and LCM for natural numbers with exampel
Answers
Answer:
The HCF of given numbers is never greater than any of the numbers. The LCM of given numbers is never less than any of the numbers. The HCF of two or more prime numbers is always 1. The LCM of two or more prime numbers is nothing but their product.
Example 1: The LCM of two numbers is 54 and their HCF is 3. If one of these numbers is 27, find the other number. Solution: As per the properties of HCF and LCF, HCF × LCM = Product of two numbers. ...
Example 2: Two numbers are said to be in the ratio 2 : 3. If their L.C.M. is 48.
Property 1
The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
LCM × HCF = Product of the Numbers
Suppose A and B are two numbers, then.
LCM (A & B) × HCF (A & B) = A × B
Example: If 3 and 8 are two numbers.
LCM (3,8) = 24
HCF (3,8) = 1
LCM (3,8) x HCF (3,8) = 24 x 1 = 24
Also, 3 x 8 = 24
Hence, proved.
Note: This property is applicable for only two numbers.
Property 2
HCF of co-prime numbers is 1. Therefore, LCM of given co-prime numbers is equal to the product of the numbers.
LCM of Co-prime Numbers = Product Of The Numbers
Example: Let us take two coprime numbers, such as 21 and 22.
LCM of 21 and 22 = 462
Product of 21 and 22 = 462
LCM (21, 22) = 21 x 22
Property 3
H.C.F. and L.C.M. of Fractions:
LCM of fractions =
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HCF of fractions =
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Example: Let us take two fractions 4/9 and 6/21.
4 and 6 are the numerators & 9 and 12 are the denominators
LCM (4, 6) = 12
HCF (4, 6) = 2
LCM (9, 21) = 63
HCF (9, 21) = 3
Now as per the formula, we can write:
LCM (4/9, 6/21) = 12/3 = 4
HCF (4/9, 6/21) = 2/63
Property 4
HCF of any two or more numbers is never greater than any of the given numbers.
Example: HCF of 4 and 8 is 4
Here, one number is 4 itself and another number 8 is greater than HCF (4, 8), i.e.,4.
Property 5
LCM of any two or more numbers is never smaller than any of the given numbers.
Example: LCM of 4 and 8 is 8 which is not smaller to any of them.
Solved Problems
Example 1: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12
Solution:
9 = 3 × 3 = 3²
12 = 2 × 2 × 3 = 2² × 3
LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36
HCF of 9 and 12 = 3
LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108
Product of 9 and 12 = 9 × 12 = 108
Hence, LCM (9 & 12) × HCF (9 & 12) = 9 × 12 = 108. Proved.
Example 2: 8 and 9 are two co-prime numbers. Using these numbers verify, LCM of Co-prime Numbers = Product Of The Numbers.
Solution: LCM and HCF of 8 and 9:
8 = 2 × 2 × 2 = 2³
9 = 3 × 3 = 3²
LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72
HCF of 8 and 9 = 1
Product of 8 and 9 = 8 × 9 = 72
Hence, LCM of co-prime numbers = Product of the numbers. Therefore, verified.