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

Answer:

smallest positive integer with 5 positive divisor is 16

smallest positive integer with 60 positive divisor is 5040

Step-by-step 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