Question

(a)What is the smallest positive interger that has exactly 45 disticnt postive divisors inlcluding 1 and itself

detailed please

Answer 1

As we know that for any integer n

n=a^p∗b^q∗c^r

where a, b, and c are prime factors of p, q, and r are their powers.

The number of factors of nn will be expressed by the formula (p+1)(q+1)(r+1).

and this will include 1 and n itself.

for 45 disticnt postive divisor with samlles positive number

we can have

n = 2^44

then number of posiive divisore = 44+1 = 45

or another case

n = 3^2 * 2^14

another case

n = 5^2 3^2 2^4

In above three case

we can see that last case has smallesr positive number that has exact 45 positive divisor

Answer: Number = 5^2 * 3^2 * 2^4 = 3600