The LCM of two numbers is 9 times their HCF.
HCF and LCM Relationship Problem uses the given condition that the LCM is nine times the HCF and their sum is 500 to determine that the HCF is 50.
The LCM of two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find their HCF.
Let the HCF of two numbers be x.
Then, LCM = 9x
According to the question,
$\mathrm{LCM}+\mathrm{HCF}=500$
$\Rightarrow 9 x+x=500$
$\Rightarrow 10 x=500$
$\Rightarrow x=50$
Hence, the HCF of two numbers is 50.
Frequently Asked Questions
Find answers to common questions.
What is the HCF when LCM is 9 times the HCF and their sum is 500?
The HCF is 50. Setting HCF = x and LCM = 9x, the equation 9x + x = 500 gives 10x = 500, so x = 50. The LCM works out to 450. This is the standard single-variable substitution method taught in CBSE Class 10 Real Numbers.
What is the formula connecting LCM and HCF of two numbers?
The key formula is LCM × HCF = Product of the two numbers. This holds strictly for two positive integers. If LCM and HCF are known along with one of the numbers, you can always find the other by dividing the product (LCM × HCF) by the known number.
Can LCM ever be less than HCF?
No, never. For any two positive integers, HCF ≤ each number ≤ LCM. The LCM is always greater than or equal to the HCF, with equality only when both numbers are identical. This is why the ratio LCM/HCF is always ≥ 1.
How do you find two numbers when HCF and LCM are known?
Knowing only the HCF and LCM is not enough to uniquely identify the two numbers — there may be multiple valid pairs. However, if you also know the sum, difference, or one of the numbers, you can use the identity LCM × HCF = a × b along with that extra condition to find both values uniquely.
Why does LCM × HCF = the product of two numbers?
Because HCF captures the common prime factors and LCM captures all prime factors (each to its highest power) across both numbers. Multiplying LCM and HCF effectively counts every prime factor exactly as many times as it appears in a × b. This identity is proved formally in CBSE Class 10 Chapter 1 (Real Numbers) using prime factorisation.