Great question!

We'll use the following formula:

If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = (5)(4)(2) = 40
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Okay, now let's answer the question!

P(selected number is divisible by 21) = (# of divisors that are divisible by 21)/(TOTAL # of divisors)


As always, we'll begin with the....

DENOMINATOR
We're given the number: (3³)(5²)(7²)
So, the number of divisors = (3+1)(2+1)(2+1) = (4)(3)(3) = 36

NUMERATOR - i.e., # of divisors that are divisible by 21
Let's factor 21 out of the given number.
(3³)(5²)(7²) = (3¹)(7¹)[(3²)(5²)(7¹)]
= 21[(3²)(5²)(7¹)]
At this point, we can see that, if we multiply 21 by any factor of [(3²)(5²)(7¹)], the product will definitely be divisible by 21
So, the question becomes "how many divisors does [(3²)(5²)(7¹)] have?"
Once again, we'll apply our nice formula.
We have (3²)(5²)(7¹)
So, the number of divisors = (2+1)(2+1)(1+1) = (3)(3)(2) = 18
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So, P(selected number is divisible by 21) = 18/36
= 1/2

Answer:

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Good catch! I've edited my response accordingly (the answer is still E though )
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Total number of factors = (3+1)*(2+1)*(2+1) = 4*3*3
Dividing the number by 21 , n = 3^2*5^2*7^1, Favorable number of factors =(2+1)*(2+1)*(1+1) = 3*3*2

Required probability = 3*3*2/(4*3*3) = 1/2

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