Question

Take the smallest odd composite number and the largest three digit prime number. What is the difference between the LCM and the HCF of these two numbers?
$
  (a){\text{ 8336}} \\
  (b){\text{ 8972}} \\
  (c){\text{ 8874}} \\
  (d){\text{ 8566}} \\
 $

Answer 1

Hint: In this question we have to find the difference between the LCM and the HCF of the smallest odd composite number and the largest three digit prime number. A composite number is one whose factorization is the product of two natural numbers except 1 and itself, and a prime number is one which is divisible by one and itself. Use these definitions to get the desired numbers, then find it’s HCF and LCM.

Complete step-by-step answer:

As we know that the composite number is a number whose factorization is a product of two natural numbers except 1 and itself.

So from the above definition of composite number we can say that the prime number can never be a composite number because the prime number is a product of 1 and itself.

So the smallest odd composite number from numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ……………………) is 9.

Because 1, 2, 3, 5 and 7 is a prime number and 4, 6 and 8 are even composite numbers.

Therefore 9 is the smallest odd composite number and it is written as multiplication of two three’s.
$ \Rightarrow 9 = 1 \times 3 \times 3$

Therefore 4 is the smallest composite number.

Now as we know, the prime number is a product of 1 and itself.

So the largest three digit prime number is 997.
$ \Rightarrow 997 = 1 \times 997$

Now we have to find out the difference between the L.C.M and the H.C.F of these two numbers.

So first find out the L.C.M of these two numbers.

Least common multiple of two numbers is to first list the prime factors of each number. Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
$ \Rightarrow L.C.M = 1 \times 3 \times 3 \times 997 = 8973$

Now find out the H.C.F of these two numbers.
As we know that the H.C.F of the numbers is the common factor of the numbers.

So as we see that the common factors of 9 and 997 is 1.
$ \Rightarrow H.C.F = 1$
So, the difference between the L.C.M and the H.C.F of these two numbers is
$ \Rightarrow L.C.M - H.C.F = 8793 - 1 = 8792$
So this is the required answer.

Hence option (B) is correct.

Note: Whenever we face such types of problems the key concept is simply to have the basic understanding of the definition of prime and composite number. LCM and HCF refers to lowest common factors and highest common factor and thus can easily be obtained after writing down the factors of a specific number. Use these concepts to get on the right track to reach the answer.