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GMAT Inequalities is a high-frequency topic in GMAT Quant, but many students struggle because the concepts behave differently from standard algebra. Understanding the right rules, patterns, and edge cases can significantly improve both speed and accuracy.
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02 Jul 2025, 21:05
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The total # of factors can be obtain by expressing the number in the prime factorization form and then multiplying 1+ the exponent of each prime factor.
example
12 = 2^2 * 3
Total factors: (2+1) (1+1) = 6
For max the number of prime factors i must express 24 in the longest multiplication form : 2 * 2 * 2 * 3
That is the same as (1+1)(1+1)(1+1)(2+1),
each parenthesis means one prime factor, so 4 is the max number of prime factors
But the question ask the range, so i need to also find the min number of prime factors.... simple.... (23+1) equals 24 and means 1 prime factor.
So the range is 4-1 = 3
example
12 = 2^2 * 3
Total factors: (2+1) (1+1) = 6
For max the number of prime factors i must express 24 in the longest multiplication form : 2 * 2 * 2 * 3
That is the same as (1+1)(1+1)(1+1)(2+1),
each parenthesis means one prime factor, so 4 is the max number of prime factors
But the question ask the range, so i need to also find the min number of prime factors.... simple.... (23+1) equals 24 and means 1 prime factor.
So the range is 4-1 = 3
02 Jul 2025, 21:05
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Bunuel
We know that the number of positive factors of a number M whose prime factorised form is \({P_1}^x* {P_2}^y*........ {P_n}^z\) is \((x+1)(y+1).....(z+1)\)
so the minimum number of prime factor could be 1, as the number could be of the form \({P_1}^{23}\)
To maximize the number of distinct prime factors, we must factor 24 as a product of as many small integers greater than 1 as possible, since each factor corresponds to an exponent plus 1. What we are basicall doing is maximizing the number of terms in:
\((x+1)(y+1).....(z+1)\)
so, 24= \(2^3*3^1\) , thus 3+1=4 maximum prime factors
now we are asked the range, which would be 4-1=3
Thus, IMO C.
02 Jul 2025, 21:13
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To find no of prime factors we do prime factorization so in this case it would be 24=3*2*2*2. Now we need to find the range of maximum possible in that we subtract min and max no of prime factors. So minimum is always 1 ie. the number itself. Range would be 4-1=3 Option C
Bunuel
