Question

Given there are three positive integers a, b and c. Their greatest common divisor is D and their least common divisor is M. Then, two of the following statements are true:
(1) The product MD cannot be less than abc.
(2) The product MD cannot be greater than abc.
(3) MD equals abc only and only if a, b and c are each prime.
(4) MD equals abc only and only if a, b and c are relatively prime in pairs.

Answer 1

Hint: First, before proceeding for this, we must know the definition of the GCD and LCM to proceed further. Then, we know that GCD stands for greatest common divisor which says that for two integers which are not zero gives the largest positive integer that divides each of the integers. Then, the definition of LCM which is the lowest common factor says that for integers gives the smallest positive integer that is divisible by both the integers. Then, by using these definitions, we get the correct options.

Complete step by step answer:
In this question, we are supposed to find two correct statements from the given options when there are three positive integers a, b, and c, and their greatest common divisor is D and their least common divisor is M.
So, before proceeding for this, we must know the definition of the GCD and LCM to proceed further.
So, we know that GCD stands for greatest common divisor which says that for two integers which are not zero gives the largest positive integer that divides each of the integers.
Similarly, the definition of LCM which is the lowest common factor says that for integers gives the smallest positive integer that is divisible by both the integers.
So, now by using these two definitions, we get the conditions for the question.
Here, firstly we will represent a, b and c in terms of their prime factors and then we know by the definition that D is the product of all the common prime factors where each factor is taken as it appears the least number of times in a, b or c.
Subsequently, we know M is the product of all the non-common prime factors where each factor is taken as often as it appears the greatest number of times in a, b or c.
Therefore, we get the condition that MD may be less than abc which is not conclusive but it is evident that MD can’t exceed abc.
So, option (2) is correct in any way.
Moreover, we know MD is equal to abc only when a, b and c are all prime numbers which give the option (3) as correct.
Hence, option (2) and (3) are correct.

Note:
Now, to solve these types of questions we need to know some of the basics of the number system. So, the required definition is a prime number which is a number that is disabled by only itself and 1 and can’t be divided with any other number.