An interesting Problem Solving question from Number Theory. You need to know all the basic properties of LCM and HCF of two numbers and the relation between LCM and HCF. If you are already not familiar with the topics, start with understanding what LCM and HCF actually denote (not what the acronyms stand for).
If the LCM of two numbers ‘a’ and ‘b’ is 1104 and their HCF is 4, which of the following MUST be true?
I. a * b = 4416
II. a and b are both divisible by 8
III. a : b = 48 : 23 or a : b = 23 : 48
A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III
Few results relating to LCM and HCF that you need to know to solve this question
Result 1: Product of two numbers is the same as the product of the LCM and HCF of those two numbers.
i.e., If the numbers are a and b, a * b = LCM (a, b) * HCF (a, b)
Result 2: Let the HCF of a and b is ‘h’ and the LCM of a and b is L.
a can be expressed as m*h and b can be expressed as n*h because h is a factor common to both the numbers.
So, a = mh and b = nh.
Note, m and n are co-prime because ‘h’ is the HCF of the two numbers. HCF of two numbers holds all factors common to both the numbers.
The LCM of a and b, l = m*n*h
i.e., the HCF of two numbers will be a factor of the LCM of the two numbers.
Data given in the question
LCM of a and b is 1104 and their HCF is 4.
Let us evaluate statement I
Using Result 1, we can deduce a * b = LCM (a, b) * HCF (a, b).
So, a * b = 1104 * 4 = 4416. Statement I is true.
Let us evaluate statement II
The HCF of a and b is 4. So, a and b will be divisible by 4.
If 8 could divide both a and b, the HCF of the two numbers would have been 8 and not 4.
So, statement II is NOT true.
Let us evaluate statement III
The LCM of the two numbers a and b, which is l = m * n * h if the HCF of the two numbers is h.
Then the ratio of the two numbers a and b will be m : n.
1104 = m * n * 4
Or m * n = 276 and m and n should be co-prime.
If m and n are 48 and 23, the product is 1104 and not 276.
So, statement III is NOT true.
Statement I alone is true. Choice A is the correct Answer.