What is the smallest positive interger that has exactly 45 disticnt postive divisors inlcluding 1 and itself
detailed please
Solution:
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
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