How Many Numbers Have Exactly 5, 6, or a Specific Number of Factors Between 1 and 100?
Introduction
When dealing with numbers between 1 and 100, and specifically understanding which numbers have an exact number of factors such as 5, 6, or more, we can significantly enhance our knowledge of number theory through a structured approach involving prime factorization.
Numbers with Exactly 5 Factors Between 1 and 100
To determine how many numbers between 1 and 100 have exactly 5 factors, we need to understand the relationship between the prime factorization of a number and its factors.
The number of factors of a number ( n ) is determined by its prime factorization. If ( n ) can be expressed as:
[ n p_1^{e_1} times p_2^{e_2} times ldots times p_k^{e_k} ]
where ( p_1, p_2, ldots, p_k ) are distinct prime factors and ( e_1, e_2, ldots, e_k ) are their respective positive integer exponents, then the total number of factors ( d(n) ) is given by:
[ d(n) (e_1 1)(e_2 1) ldots (e_k 1) ]
For a number to have exactly 5 factors, the product ( (e_1 1)(e_2 1) ldots (e_k 1) ) must equal 5. The only way to achieve this is if ( n ) is in the form ( p^4 ) where ( p ) is a prime number, because:
[ 5 5 Rightarrow (e_1 1) 5 Rightarrow e_1 4 ]
Thus, the numbers with exactly 5 factors must be in the form ( p^4 ) where ( p ) is a prime number.
Now, let's find the prime numbers whose fourth powers are less than or equal to 100:
[ 2^4 16 ] [ 3^4 81 ] [ 5^4 625 ] (too large)The valid numbers within the range of 1 to 100 are 16 and 81.
Therefore, the numbers between 1 and 100 that have exactly 5 factors are:
16 81In conclusion, there are 2 numbers between 1 and 100 that have exactly 5 factors.
Numbers with Exactly 6 Factors Between 1 and 100
A number has exactly 6 factors if it is a square of a prime number. The prime factorization of such a number would be ( p_1^2 ) where ( p_1 ) is a prime number.
Between 1 and 100, there are prime numbers whose squares are less than or equal to 100. These primes are 2, 3, 5, and 7. Therefore, there are 4 such numbers: ( 2^2, 3^2, 5^2, ) and ( 7^2 ).
So there are four numbers between 1 and 100 that have exactly 6 factors.
Numbers with 6 Factors: Square Numbers
Numbers with exactly 6 factors are square numbers. Between 1 and 100, there are 10 square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
How Many Multiples of 5 Between 1 and 100 (Not Including 100)
To find the number of multiples of 5 between 1 and 100 (not including 100), we can use the following method:
100 ÷ 5 20 —1 19 Do not include 100 95 ÷ 5 19So, the multiples from 1 to 100 (not including 100) are as follows:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95From the above, we can see that there are 19 such multiples.