For a given two natural numbers the product of their Hcf and Lcm is always equal to the product of that two given number numbers but this result is not true in case of three or more numbers. In other words, relation between hcf and lcm is the product of two given natural number is equal to the product of that natural number.
For any two numbers ‘a’ and ‘b’ we always have,
(a × b) = Product of their hcf and lcm
Caution:- The above result is only valid for two natural numbers not three or more.
Hence, the formula for relation between hcf and lcm for two numbers is (a × b) = Product of their hcf and lcm.
Before we discuss about relation between hcf and lcm in more detailed. we should know the meaning of HCF and LCM. so lets start with HCF.
What is hcf?
Hcf stands for highest common factor. hcf is defined as the highest number having common to all the number’s factor and divides all the number. for example lets take two natural number 6 and 10. the factor of 6 is (2,3) and 10 is (2,5) so here 2 is the highest common factor in both the numbers 6 and 10. so 2 is the hcf of (6,10).
Examples:- find the hcf of 60, 84, and 108.using prime factor method.
60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 ×7
108 = 2 × 2 × 3 × 3 ×3
hcf of (60, 84 and 108) = 2 × 2 × 3 = 12
What is Lcm?
Lcm stands for lowest common factor. lcm is defined as the lowest number that is common or non-common and multiple of all the given numbers. for example lets take two numbers 60 and 84. so the factor of 60 is (2 × 2 × 3 × 5) and 84 is (2 × 2 × 3 × 7). so the lowest common or non-common and multiple of all the numbers is (2 × 2 × 3 × 5 × 7). so the lcm of (60, 84) is 420.
Verify the relation between lcm and hcf with one example
so lets, verify the relation between hcf and lcm of two numbers with one example.
Example:- lets two natural numbers 10 and 15.
lets find prime factor of 10 and 15.
10 = 2 × 5
15 = 5 × 3
Hcf of (10, 15) = 5
Lcm of (10, 15) = 5 × 2 × 3 = 30
hence, as we know that the relation between hcf and lcm is (a × b) = Product of their hcf and lcm.
(a × b) = Product of their hcf and lcm
(10 × 15) = Hcf of (10, 15) × Lcm of (10, 15) = 150
Hence, the relation between hcf and lcm of two numbers is verified.
Relation between hcf and lcm questions
Q.1 The hcf and lcm of two numbers is 27 and their LCM is 162. If one of the numbers is 54. find the other.
Q.2 Ravi and Sikha drive around a circular sports field. Ravi takes 16 minutes to take one round, while Sikha completes the round in 20 minutes. If both start at the same point, at the same time and go in the same direction, after how much time will they meet at the starting point.
Relation between hcf and lcm of three numbers
The relation between hcf and lcm of three numbers is given by the prime factor. In other words, If p, q and r, are the three numbers then the relation between is
LCM (p, q, r) = pqr × HCF (p, q, r) divided by HCF (p,q) × HCF (q, r) × HCF (r, p)
For example we use a technique to find the HCF and LCM both together, and that technique called prime factor. lets do that,
Here we use three given numbers 70, 80, and 120.
prime factor of,
70 = 2 × 5 × 7
80 = 16 × 5
120 = 8 × 3 × 5
As we know that,
HCF is he lest powered of shared factor.
LCM is the highest powered of each factor.
Here lest powered shared factor is 2 and 5. in each number.
HCF of (70, 80 and 120) = 2 × 5 = 10
LCM of (70, 80 and 120) = 1680