Question

P1. (3+7 points) What is the smallest positive integer with precisely 5 positive divisors? What is the smallest positive integer with precisely 60 positive divisors? Show your work and reasoning.

Answer 1

smallest positive integer with 5 positive divisor is 16

smallest positive integer with 60 positive divisor is 5040

Explanation:

given data

precisely positive divisors = 5

precisely positive divisors = 60

solution

we take here
`a^(x) *b^(y) *c^(z)` is express as

= (x+1) × (y+1) × (z+1)

so put here now x is 4

and z = y = 0 and a is least integer more than 1 it will be 2

and b and c ≥ 1

and
`x^(0)` is = 1

so
`a^(x) *b^(y) *c^(z)` is

`a^(4)` is =
`2^(4)` = 16

so smallest positive integer with 5 positive divisor is 16

and

same like 60 positive divisors

dn = ( a1+1 ) × ( a2+1 ) × ( a3+1 ) ............ ( an+1 )

n =
`p1^(a1) * p2^(a2) * p3^(a3) * ............ pn^(an) *`

n = 7 × 5 × 3² ×
`2^(4)`

n = 5040

smallest positive integer with 60 positive divisor is 5040