Question

Let a, b, and c be positive integers that form a geometric sequence. Given that a has 3 factors and c has 9 factors, how many factors can b have, at most?
Answer choices: 14, 9, 6, 8, 10.
DO NOT ATTEMPT IF YOUR NOT 98% SURE

Answer 1

Answer:

9

Step-by-step explanation:

it’s clear that a ,b and c are in geometric progression.

Then we have :

a

b = q×a

c = q²×a

also , b² = a×c

………………………

NOTE : b² and b have the same factors

also ,since b² = a×c then b and a×c

have the same number of factors.

_______________________________

Since c = q²×a then any factor of a is also a factor of c

Which means the number of factors of a×c

is equal to the number of factors of c.

Therefore b at most has the same number of factors as c.

Conclusion:

at most b has 9 factors.

……………………

As a side note :

You can take as example the numbers 3 , 30 , 300.

3 ——-> 1 factor

30 = 3×5×2 ——-> 3 factors

300 = 3 × 5² × 2² ——-> 3 factors