An interesting GMAT Problem Solving question from Number Properties & Number Theory. It tests your understanding of LCM and HCF and certain key results regarding LCM and HCF of two or more numbers.
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
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)
Note: a * b * c NEED NOT be equal to LCM(a, b, c) * HCF(a, b, c).
This rule works for 2 numbers, irrespective of whether the numbers are both integers, both fractions, one fraction and the other an integer.
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.
a = mh and b = nh.
Note, m and n are co-prime (have no factor in common) because ‘h’ is the HCF of the two numbers. HCF of two numbers holds all factors common to both the numbers.Hence, we can deduce that the LCM (a, b), L = m*n*hi.e., the HCF of two numbers will be a factor of the LCM of the two numbers.
Data given in the question stem
Statement I: a * b = 4416
Result 1 states that a * b = LCM (a, b) * HCF (a, b).
So, a * b = 1104 * 4 = 4416.
Statement II: a and b are both divisible by 8
The HCF of a and b is 4. So, the largest number that could divide both a and b is 4.
If 8 could divide both a and b, the largest number that could divide both would have been 8.
Consequently, the HCF of the two numbers would have been 8 and not 4.
So, statement II is NOT true.
Statement III: a : b = 48 : 23 or a : b = 23 : 48
Where a = mh and b = nh and m and n are co-prime.
i.e., we have to determine whether m : n = 48 : 23 or 23 : 48.
Because L = m * n * h, 1104 = m * n * 4
Note: m and n are co-prime.
If m and n are 48 and 23 or vice versa, m * n = 1104 and not 276.
Choice A. Statement I alone is true.