Respuesta :
Answer:
[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:
[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]
Step-by-step explanation:
From the info given by the problem we need an integer defined as the smallest positive integer that is a multiple of 75 and have 75 positive integral divisors, and we are assuming that 1 is one possible divisor.
Th first step is find the prime factorization for the number 75 and we see that
[tex]75=3 5^2[/tex]
And we know that 3 =2+1 and 5=3+2 and if we replace we got:
[tex] 75 = (2+1)(4+1)^2 = (2+1)(4+1)(4+1)[/tex]
And in order to find 75 integral divisors we need to satisify this condition:
[tex]n= a^{r_1 -1}_1 a^{r_2 -1}_2 *......[/tex] such that [tex]a_1 *a_2*....=75[/tex]
For this case we have two prime factors important 3 and 5. And if we want to minimize n we can use a prime factor like 2. The least common denominator between 2 and 4 is LCM(2,4) =4. So then the need to have the prime factors 2 and 3 elevated at 4 in order to satisfy the condition required, and since 5 is the highest value we need to put the same exponent.
And then the value for n would be given by:
[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:
[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]
The smallest positive integer that is a multiple of 75 is 32400 and
integral divisors are 432.
Positive integer
Positive integers are the whole number that is greater than zero and do not include decimal or fraction values.
Find the smallest positive integer that is a multiple of 75 that has exactly 75 positive integral divisors.
How to calculate?
We know that
[tex]75 = 3*5^{2}[/tex]
So the value of n which has 75 divisors then the multiplication of power of prime factor should be 75. The formula is given by
[tex]n = 2^{x-1} *3^{y-1} *5^{z-1}[/tex]
then the multiplication of x, y, and z must be 75.
x, y, and z are 5, 5, and 3 will be the values.
[tex]n = 2^{4} *3^{4} *5^{2} = 32400[/tex]
And it is divisible by 75 also.
[tex]\dfrac{n}{3*5^{2} } = \dfrac{2^{4} *3^{4} *5^{2}}{3*5^{2} } = 432[/tex]
Thus, the smallest positive integer that is a multiple of 75 is 32400 and integral divisors are 432.
More about the Integer link is given below.
https://brainly.com/question/1768254
