Question

What is the smallest number that has exactly 8 factors?

• A. 2⁷
• B. 24
• C. 54
• D. 18

Answer 1

The correct option is B

24

Number of the form $p^a{q}^b{r}^c$ (where p, q and r are primes) has (a + 1) (b + 1) (c + 1) factors.
So, if some number has 8 factors (a + 1)(b + 1)(c + 1) should be 8.
8 can be written as 8 or $4\times 2$ or $2\times 2\times 2$.
If a number has to have 8 factors, it should be in one of the following forms,
$p^7⟹7+1=8$ factors
$p^3{q}^1⟹4\times 2=8$ factors
$pqr⟹2\times 2\times 2=8$ factors
Let us deduce the smallest possible number in each form.
$p^7⟹2^7$
$p^3{q}^1⟹2^3\times 3=24$
$pqr⟹2\times 3\times 5=30$

Therefore, smallest number that has exactly 8 factors = 24.