Question

M and N are two natural numbers such that $M = 51$ . LCM of M and N is 255. What is/are the HCF of M and N?
A) 3
B) 5
C) 1
D) 13

Answer 1

Hint:
We can use the relation that the product of two numbers will be equal to the product of their LCM and HCF to get the value of HCF from N. Then we can find the possible values of N that give the given LCM. Then we can find the HCF in each case using the relation we obtained in the $1^{\text{st}}$ step.

Complete step by step solution:
We are given that M and N are two natural numbers.
 $M = 51$
 $LCM\left( {M,N} \right) = 255$
We know that the product of 2 numbers will be equal to the product of their LCM and HCF.
So, we can write,
 $M \times N = LCM \times HCF$
On substituting the values, we get,
 $ \Rightarrow 51 \times N = 255 \times HCF$
We can rewrite this equation as,
 \[ \Rightarrow HCF = \dfrac{{51 \times N}}{{255}}\]
On simplification, we get,
 \[ \Rightarrow HCF = \dfrac{N}{5}\] … (1)
Now we can find all the possible values of N. For that we can factorise the given LCM and write it as a product of the prime factors.
On factorising, we get,
 $255 = 5 \times 51$
On factorising 51, we get,
 $ \Rightarrow 255 = 5 \times 3 \times 17$
So, we get LCM as 255 when N is $5 \times 3 \times 17 = 255$ , $5 \times 3 = 15$ , $5 \times 17 = 185$ or 5.
When $N = 225$ , HCF of M and N is given by (1),
 \[ \Rightarrow HCF = \dfrac{{255}}{5}\]
So, we have,
 \[ \Rightarrow HCF = 51\]
When $N = 15$ , HCF of M and N is given by equation (1),
 \[ \Rightarrow HCF = \dfrac{{15}}{5}\]
So, we have,
 \[ \Rightarrow HCF = 3\]
When $N = 185$ , HCF of M and N is given by equation (1),
 \[ \Rightarrow HCF = \dfrac{{185}}{5}\]
So, we have,
 \[ \Rightarrow HCF = 17\]
When $N = 5$, HCF of M and N is given by,
 \[ \Rightarrow HCF = \dfrac{5}{5}\]
So, we have,
 \[ \Rightarrow HCF = 1\]
So, the possible values of the HCF of M and N are 51, 3, 17 and 1.
Out of these, only 3 and 1 are given in the options.

So, the correct answers are options A and C which are 3 and 1.

Note:
We must find all the possible values of N and its corresponding HCF. The possible values of N are found by multiplying 5 with each of the factors of M. Other values are not possible because, if N is not a multiple of 5, it will become the factor of M and their LCM will be M and will not be equal to 255.
LCM or least common multiple of 2 two numbers is the smallest possible number which is a multiple of both the numbers.
HCF or highest common multiple of 2 numbers is the highest of all the common factors of the two numbers.