Question

The difference between the LCM and HCF of the natural numbers a and b is 57. What is the minimum value of a + b?
A. 22
B. 27
C. 31
D. 58

Answer 1

Hint: As the difference between lcm and hcf is given, we will let hcf be k and use it to find the sum of natural numbers a and b.

Complete step-by-step answer:
Now, let hcf be k. So, the natural numbers are
a = kx, b = ky, where x and y are co – primes.
Now, therefore by the definition of lcm, lcm is least common multiple of two numbers, so lcm of a and b is lcm (a, b) = kxy
Now, the difference of lcm and hcf is equal to 57.
Therefore, kxy – k = 57
k (xy – 1) = 57
xy – 1 = \[\dfrac{{57}}{{\text{k}}}\]
Now, we have to put only those values of k in the above expression, which will make the right – hand side an integer, because the product of two integers is always an integer.
Putting k = 1, we get
xy – 1 = 57
xy = 58
therefore, x = 2, y = 29
So, a = kx = 2, b = ky = 29
Therefore, a + b = 2 + 29 = 31
Now, putting k = 3, we get
xy – 1 = 19
xy = 20
x = 4, y = 5
so, a = kx = 12, b = ky = 15
Therefore, a + b = 27
So, the minimum value of a + b = 27
So, option (B) is correct.

Note: When we come up with such types of questions, we have to let HCF as a variable, then find the value of two numbers by finding the LCM. Then we will add the numbers to get the desired answer. The type of value of HCF to be put in the condition given in the question should be checked. Many students start putting the value of HCF from 1 which leads them to incorrect answers, because putting such values will give fractional values which is not possible because both the given numbers are natural.