James' age is between that of Gwen and Lucille and is closer to Gwen's

Lucille is less than 70 years old, and her age In years has exactly three prime factors

i think Lucille age has exactly 3 prime factors ( e.g. : 66 - 3*2*11 ) but nothing is mentioned about age of James

please correct me if i am wrong Bunuel IanStewart

Yes, Lucille's age has three prime factors, so could be 66, or 42, among other possibilities (it could be 30 or 60 too).

That's what my solution you have quoted says (I'm not sure if you think it says something else?).

Given:
i. G < J << L ,
ii. James' age is closer to Gwen's age than Lucile's.
iii. 30 <= G <= 40

Pt.1 suggests James' age is less than that of Gwen's. Not possible.
Pt.3 suggests James' age is closer to Lucile's age than Gwen's. Not possible.

Since the answer options suggest at least one the options is true. We can say that James' age can possibly be 37. Option suggesting only pt 2 is correct.

Ans: B

Solution

Given

• Gwen is between 30 and 40 years old, inclusive.
• Lucille is less than 70 years old, and her age in years has exactly three prime factors. Let us find out all the possibilities.

To Find

• Which of the options is the possible age of James.

Approach and Working Out
Let us assume the ages for Gwen, Lucile, and James are G, L, and J respectively.

Then, G < J < L

The least age possible for G is 30 and James is elder to Gwen.

• Hence, J must be more than 30.

The highest age possible for G is 40 and the highest age possible for Lucile is 69 (Less than 70).

• The Highest possible age for James is less than \(\frac{(40 + 69)}{2}\) or less than 55. (As James’ age is closer to that of Gwen.

We can conclude that the age of James must lie in the range 30 to 55, exclusive.

Only 37 years (II) is correct.

Correct Answer: Option B

The prime factors whose product are to be less than 70 shall have at max 2,3,5, 7, 11. As luccile is less than 70 (5 x 5 x 5 and 3 x 5 x 5 cannot be considered).

Thus, the minimum age of lucile can be 2 x 2 x 2 = 8 and maximum can be 2 x 3 x 3 = 54

As James is between Glen and Lucile :

Case 1 : James 22 - Lucile 8, Glen 30-40 - James can be anything from 29 to 9 years old, thus 22 is possible.

Case 2 : James 37 - Lucile 54, Glen 30 - 40 - James can be 37, when Glen is 30 years old. Hence 37 is possible.

Case 3 : James 65 - Lucile 66 - Glen 30 - 40 Possible but James is not closer to Glen. Hence cannot be the answer.
Please correct me if i am wrong.

There are a few issues in your solution. Lucille's maximum age is 66 (2*3*11), not 54 (it would be 68 (2^2*17) if you think you should count the '2' twice, but you shouldn't) and I think you left out a '3' when writing out 54 as a product.

In your Case 1, James must be at least 20 in order to be closer in age to Glen than to Lucile (you find James' minimum age by making Glen as young as possible, so by making him 30).

But most importantly, if a number has three prime factors, those prime factors are distinct. The number 8 has only one prime factor, 2. There's no reason to count the '2' three times, just as if you had a bag with three red marbles in it, and were asked "how many colours are in the bag?" you would not count the colour "red" three times -- there is only one colour in the bag. I can't count how prep company books I've seen that either get this wrong or make it into a confusing point, so it's perfectly understandable that many test takers are also confused about it.

L's age has 3 prime factors - means at least 2*3*5 = 30 years old and less than 70 means - 2*5*7=70 not possible. hence, L's age is 30 years.

G's age is between 30 and 40 and James' age is between both of them and closer to G's age.

Hence only II fits in. Answer B.

Solution:

We see that James’ age can range anywhere between 30 and 70, so we see that his age can’t be 22. His age could be 37 or 65 unless Lucille is less than 38 years old. We know that she is less than 70 years old, but we need to determine the possible values for her age). Since her age has exactly three prime factors, she can be 2 x 3 x 7 = 42, 3^2 x 5 = 45, 2 x 5^2 = 50, or 3^2 x 7 = 63 years old (notice that we used only those prime products that yielded a value between 30 and 70). Since none of these numbers are greater than 65, we see that James can’t be 65 years old. Therefore, James could only be 37 years old (if Lucille is 42 and Gwen is, for example, 35).

Answer: B

The numbers less than 70 with three prime factors are 30, 42, 60 and 66. The numbers 45, 50 and 63 have only two prime factors. Because 66 is greater than 65 and has three prime divisors, you can't solve the problem without using the information that James' age is closer to Gwen's age than it is to Lucille's.

To confirm, when GMAT says 3 prime factors we must assume distinct prime factors?

The GMAT either always or almost always says "distinct prime factors" anyway (the question in this thread is not an official question), but if a question did only say "x has 3 prime factors", that means the same thing as "x has three distinct prime factors". If you think of a number like 12 = (2^2)(3), it is divisible by precisely two primes, 2 and 3, so it has two prime divisors (or factors). There's no logical reason to count the '2' twice just because it has an exponent.

If you think of similar sentences, like "the positive integer x has 12 divisors" or "the positive integer x has 8 even divisors", we always mean "distinct divisors" or "distinct even divisors". We don't double-count any of the divisors, even if they might have an exponent when you break x down - for example, when counting the divisors of 48, you don't count "4" twice just because (4)(4)(3) = (4^2)(3) = 48. We only count the '4' once.

There's a historical reason so many prep books seem to get this point wrong (or at least I suspect this is the reason). There is one very old official GMAT question that asks about something known in Number Theory as the "length" of a number. When you compute "length", you do double-count repeated primes - the length of a number is just how many primes (not necessarily distinct) you need to multiply together to make that number. But that question defines "length" in the question itself, (you do not need to learn what it is) and explicitly tells you to count prime factors even when they're not distinct. It's the only official question I've ever seen where you do that, but I've seen many GMAT books and forum discussions over the last ten years that claim you're supposed to count repeated primes when asked to count "prime factors", and that's just not the case.

But as I said above, it's very unlikely you'll need to worry about this on the actual test!

Evaluate Each Option:

I. 22: Too low; James must be older than Gwen, so not possible.

II. 37: Fits. If Gwen is 36 and Lucille is 42 (the closer 3 prime factor number), James’ age of 37 is closer to Gwen and fits the age range.

III. 65: Too high; James would be closer to Lucille, violating the "closer to Gwen" condition.

Answer: The only valid option is B (37).

the question didn't say distinct prime factors, which made me answer it very differently.

Should this read "distinct prime factors"?

Gwen is 31
Lucille is 8 (2^3)

I is correct

Your doubt has already been addressed in the thread above. Please review.