Question
What is the smallest number that has exactly 8 factors?
Answer: Option B
:
B
Number of the form paqbrc (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 4×2 or 2×2×2.
If a number has to have 8 factors, it should be in one of the following forms,
p7⟹7+1=8 factors
p3q1⟹4×2=8 factors
pqr⟹2×2×2=8 factors
Let us deduce the smallest possible number in each form.
p7⟹27
p3q1⟹23×3=24
pqr⟹2×3×5=30
Therefore, smallest number that has exactly 8 factors = 24.
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:
B
Number of the form paqbrc (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 4×2 or 2×2×2.
If a number has to have 8 factors, it should be in one of the following forms,
p7⟹7+1=8 factors
p3q1⟹4×2=8 factors
pqr⟹2×2×2=8 factors
Let us deduce the smallest possible number in each form.
p7⟹27
p3q1⟹23×3=24
pqr⟹2×3×5=30
Therefore, smallest number that has exactly 8 factors = 24.
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