If 100

Question

If 100 < x^2 < 225, what is the greatest possible number of unique positive factors of integer x?

A. 3
B. 4
C. 6
D. 12
E. 15

kungfury42 · Answer

100 < x² < 225
Implies 10 < x < 15

Since x is an integer, the only values it can take are {11, 12, 13, 14}

Of these 11 and 13 are prime, meaning they have only 2 unique factors, 1 and themselves. Between 12 and 14, 12 has more factors (6) than 14 (4).

Factors of 12 = {1, 2, 3, 4, 6, 12}
Factors of 14 = {1, 2, 7, 14}

Hence, an integer x such that 100 < x² < 225 can have a maximum of 6 unique factors, option C.

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Kinshook · Answer

Asked: If 100 < x^2 < 225, what is the greatest possible number of unique factors of integer x?

x = 11 or 12 or 13 or 14.
If x=11; unique factors = 2
If x=12 = 2^2*3; unique factors = 3*2 =6
If x=13; unique factors = 2
If x=14=2*7; unique factors = 2*2=4

Greatest number of unique factors = 6

IMO C

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CareerLauncherMBA · Answer

Taking square root, hence
10 < x < 15

Therefore, x could be 11, 12, 13 or 14
12 and 14 are composite numbers

When x = 12 = 2^2 * 3, number of factors are 3 * 2 = 6
When x = 14 = 2 * 7, number of factors are 2 * 2 = 4

Greatest number of unique factors of x is 6

IMO C

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Bunuel

If 100 < x^2 < 225, what is the greatest possible number of unique positive factors of integer x?

A. 3
B. 4
C. 6
D. 12
E. 15

shivampaul · Answer

Why are we not considering -11 to -14 for this?

Bunuel · Answer

We are not ignoring them. They just do not change the count.

If x = -12, its  positive factors  are the same as those of 12:

1, 2, 3, 4, 6, 12

So -12 and 12 give the same maximum count: 6.

egmat · Answer

Hi shivampaul,

Good catch. You're right that the posted solutions jumped straight to "10 < x < 15" and quietly dropped the negative side. Let me address exactly that.

Starting from  100 < x2 < 225 , taking the square root gives  10 < |x| < 15  (not just x). So the full list of integers is actually:

-  ±11, ±12, ±13, ±14

So you're correct - the negatives  are  valid values of x. The reason the solutions still land on the same answer is what happens when you count factors.

Why the negatives don't change the count

On the GMAT, "number of factors" is counted using the  positive divisors . A number and its negative share the exact same set of factors:

- Factors of  12 : {1, 2, 3, 4, 6, 12} -  6 
- Factors of  -12 : same set {1, 2, 3, 4, 6, 12} -  6

So  -12  gives the same  6  factors as  12 ,  -14  the same  4  as  14 , and so on. Including the negatives adds no new possibilities and no larger count - the maximum is still  6 , from x =  ±12 . That's why the answer stays  C .

So your instinct to include the negatives is completely valid; they just don't beat what  12  already gives you.

Quick way to lock it in:  think of  -6 . Its factors are 1, 2, 3, 6 - exactly the factors of  6 . The sign of the number never adds or removes factors, so once you've checked the positive value, the negative is already covered.

Answer: C

If 100 < x^2 < 225, what is the greatest possible number of unique fac

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