Question

Can two numbers have $ 15 $ as their HCF and $ 175 $ as their LCM? Give reasons.

Answer 1

Hint: HCF is the greatest or the largest common factor between two or more given numbers whereas LCM is the least or the smallest number with which the given numbers are exactly divisible. It is also known as the least common divisor. LCM can be expressed as the product of constant and HCF. Here we will find the correlation between the LCM and HCF and will find the required solution accordingly.

Complete step-by-step answer:
LCM can be expressed as the product of highest power of each factor involved in the numbers.
HCF can be expressed as the smallest power of each common factor.
We can conclude that the LCM is always a multiple of HCF and any constant.
i.e. LCM $ = K \times HCF $
“K” is any constant number.
We are given that –
HCF $ = 15 $ and
LCM $ = 175 $
Place the value in the above equation –
 $ 175 = K \times 15 $
Make “K” the subject –
 $ K = \dfrac{{175}}{{15}} $
Division
 $ \Rightarrow K = 11.667 $
The “K” should be any natural number. The value of K is in decimal point and so in this case it is not possible.
So in this case LCM $ \ne K \times HCF $
In other words No, two numbers can't have 15 as their HCF and 175 as LCM because HCF of the numbers must be a factor of the LCM. Since 15 is not a factor of 175, it is not possible.

Note: Always remember that LCM is the highest number compared to HCF for two given numbers. Also, LCM contains the value of the HCF for the two given values. HCF (Highest common factor is also known as the greatest common factor. LCM (Least common multiple) is also known as the least common divisor.