Question

A) What is the smallest number that is a product of three different prime numbers ?
B) The number \[1001\] is the product of three prime numbers . One of them is \[13\] . What are the other two ?

Answer 1

Hint: We have to find the smallest number that is a product of three different prime numbers and also we have to find the other two prime factors of \[1001\] other than \[13\] which is one of the three prime factors of \[1001\] . We solve this question using the concept of prime numbers . Solving the first part , we will take the three smallest prime numbers and then multiply them to find the product of the terms which will give us the smallest number that will be the product of three different prime numbers . For solving the second part , we would first divide the given number by dividing it by the one prime factor given and then we will split the result of the division into its prime factors .

Complete step-by-step solution:
Given :
\[A)\] The smallest number which is the product of three different prime numbers :
Using the knowledge of prime numbers we can determine the three different smallest prime numbers .
The three smallest prime numbers are stated as : \[2\] , \[3\] and \[5\]
These three are the smallest prime numbers .
\[The{\text{ }}product{\text{ }}of{\text{ }}the{\text{ }}three{\text{ }}prime{\text{ }}numbers = {\text{2}} \times 3 \times 5\]
\[The{\text{ }}product{\text{ }}of{\text{ }}the{\text{ }}three{\text{ }}prime{\text{ }}numbers = 30\]
Hence , the smallest number that is a product of three different prime numbers is \[30\] .
\[B)\] The number \[1001\] is the product of three prime numbers . One of them is \[13\] then the other two are :
We are given that the number is \[1001\] . As one of the factors is \[13\] , then for finding the other two factors we will divide the 1001 by \[13\] for finding the other two factors .
The number obtained after the division of \[1001\] by \[13\] given as :
\[\dfrac{{1001}}{{13}} = 77\]
Now , we will have to find the prime factors of \[77\] .
The prime factorisation of \[77\] is as follows :
\[77 = 7 \times 11\]
The other two prime factors of the original number \[1001\] are \[7\] and \[11\] .
Hence , the three prime factors of the number \[1001\] are \[7\] , \[11\] and \[13\] .

Note: Each and every number can be represented in terms of its prime factors . The smallest prime number of the series of numbers is \[2\] . The smallest odd prime number is \[3\] . \[1\] is the factor of each and every number whether it’s a natural number , whole number , integer numbers.