Question

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

Answer 1

Hint: To find a solution to the given problem we use the fact that LCM of numbers is always divisible by numbers and also by HCF of the given numbers. If not then given HCF, LCM combination is not possible.

Complete step-by-step answer:
Given,
HCF of two numbers is $ 15 $
and
LCM of two numbers is $ 175 $
Let two numbers are ‘a’ and ‘b’.
Since, we know that HCF of numbers is the highest number that can be taken from given numbers.
Hence, HCF is a divisor of both numbers as well as product of numbers.
Also, we know that LCM of numbers is the least number which is always divisible by given numbers.
Which implies that given numbers are factors of LCM.
But we know that HCF divides its numbers.
So, from above we conclude that HCF of numbers will also divide LCM of same numbers.
Or
We can say that LCM of numbers is some multiple of their HCF.
i.e.
 $
  LCM\,\,of\,\,given\,\,numbers = K \times \,\,HCF\,\,of\,\,given\,\,numbers \\
  \dfrac{{LCM\,\,of\,\,given\,\,numbers}}{{HCF\,\,of\,\,given\,\,numbers}} = K \;
  $
If K is either negative or in fraction then we can say that HCF and LCM are not of the same numbers.
Now, from given HCF = $ 15 $ and LCM = $ 175 $
Clearly, $ \dfrac{{175}}{{15}} = \dfrac{{35}}{3} $ in fraction.
Hence, we can say $ 15 $ and $ 175 $ can’t be the HCF and LCM of two numbers.

Note: We know that HCF of numbers is always factors of numbers and numbers are factors of LCM. Therefore, we can say that HCF is a factor of LCM also, using this we can directly check if HCF exactly divides LCM then combination is possible then otherwise given pair of HCF and LCM are not possible.