Identifying the Smallest Positive Integers with Exactly 12 Divisors
Understanding the prime factorization of a number and how it relates to the number of divisors can be a fascinating exploration in number theory. In this article, we will delve into finding the smallest positive integers that have exactly 12 divisors. We will use the formula for the number of divisors and explore different prime factorizations to determine such integers.
The Formula for the Number of Divisors
The number of divisors ( d(n) ) of a number ( n ) with prime factorization ( n p_1^{e_1} times p_2^{e_2} times ldots times p_k^{e_k} ) is given by:
[ d(n) (e_1 1) times (e_2 1) times ldots times (e_k 1) ]To have exactly 12 divisors, ( d(n) 12 ). Therefore, we need to consider the different factorizations of 12:
12 12, which corresponds to ( p_1^{11} ) 12 6 times; 2, which corresponds to ( p_1^5 times p_2^1 ) 12 4 times; 3, which corresponds to ( p_1^3 times p_2^2 ) 12 3 times; 2 times; 2, which corresponds to ( p_1^2 times p_2^1 times p_3^1 )Exploring Each Case
Case 1: ( p_1^{11} )
The smallest integer for this case is ( 2^{11} 2048 ).
Case 2: ( p_1^5 times p_2^1 )
The smallest integer for this case is ( 2^5 times 3^1 32 times 3 96 ).
Case 3: ( p_1^3 times p_2^2 )
The smallest integer for this case is ( 2^3 times 3^2 8 times 9 72 ).
Case 4: ( p_1^2 times p_2^1 times p_3^1 )
The smallest integer for this case is ( 2^2 times 3^1 times 5^1 4 times 3 times 5 60 ).
Summary and Results
By comparing the results from each case, we find:
From Case 1: 2048 From Case 2: 96 From Case 3: 72 From Case 4: 60Thus, the smallest positive integers with exactly 12 divisors are:
60 72 96 2048The smallest positive integer with exactly 12 divisors is 60.
Conclusion
Understanding how prime factorization leads to the number of divisors can help in solving various number theory problems. This exploration has shown that the smallest number with exactly 12 divisors is 60, and other options like 72, 96, and 2048 are also valid but larger.