Exploring 3-Digit Numbers with Exactly 5 Factors: A Comprehensive Guide

In this article, we delve into the fascinating world of 3-digit numbers that have exactly 5 factors. We will explore the underlying mathematical concepts, including prime factorization, to identify the numbers that meet this unique criterion. This guide is perfect for students, mathematicians, and anyone interested in the intricacies of number theory.

Introduction

Understanding the concept of factors and how they are determined by a number's prime factorization is key to finding such numbers. In this article, we will walk through a step-by-step process to identify all 3-digit numbers that have exactly 5 factors. We will also explore how the J programming language can be used to solve similar problems and list the numbers.

Understanding Prime Factorization and Factors

A number ( n ) can be expressed in the form:

where ( p_i ) are distinct prime numbers and ( e_i ) are their respective positive integer exponents. The total number of factors ( dn ) of ( n ) can be calculated using the formula:

( dn (e_1 1) times (e_2 1) times ldots times (e_k 1) )

To have exactly 5 factors, ( dn 5 ). The number 5 can only be expressed as a product of natural numbers in one way: ( 5 5 times 1 ). This means ( n ) must be of the form:

( n p^4 )

where ( p ) is a prime number. This is because if ( n p^4 ), then the number of factors is ( 4 1 5 ).

Identifying 3-Digit Numbers with Exactly 5 Factors

A 3-digit number ranges from 100 to 999. We need to find the prime numbers ( p ) such that:

( 100 leq p^4 leq 999 )

For the lower bound:

( p^4 geq 100 ) implies ( p geq 100^{1/4} 10^{1/2} 10 )

Thus, ( p geq 10 ).

For the upper bound:

( p^4 leq 999 ) implies ( p leq 999^{1/4} approx 5.623 )

Thus, ( p ) must be less than or equal to 5.

The only prime numbers between 10 and 5.623 are: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 However, since ( p ) must be less than or equal to 5, we only consider the prime numbers ( p 2 ) and ( p 3 ):

For ( p 2 ): 2^4 16 quad text{not a 3-digit number} For ( p 3 ): 3^4 81 quad text{not a 3-digit number} For ( p 5 ): 5^4 625 quad text{3-digit number}

Finally, the only 3-digit number with exactly 5 factors is:

625 5^4

Conclusion

Thus, there is 1 three-digit number that has exactly 5 factors: 625.

Using the J Programming Language to Solve Similar Problems

The J programming language can be used to solve similar problems and list the numbers. For instance:

Given the query: (5/), the answer is 72.

To list these 3-digit numbers:

4 18 a~5/

The list includes the following numbers: 108, 112, 120, 162, 168, 176, 180, 200, 208, 243, 252, 264, 270, 272, 280, 300, 304, 312, 368, 378, 392, 396, 405, 408, 420, 440, 450, 456, 464, 468, 496, 500, 520, 552, 567, 588, 592, 594, 612, 616, 630, 656, 660, 675, 680, 684, 688, 696, 700, 702, 728, 744, 750, 752, 760, 780, 828, 848, 882, 888, 891, 918, 920, 924, 944, 945, 952, 968, 976, 980, 984, 990