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To find the number of factors, write the number in prime factorization form, add one to each power, and multiply. For example, 2100 = 2² × 3¹ × 5² × 7¹. Adding one and multiplying gives (2+1)(1+1)(2+1)(1+1) = 36 factors.
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In your GMAT prep, learning to solve factor-based questions is an important part of building confidence. Rather than listing factors one by one, which can be slow and prone to errors, a smarter approach lies in prime factorization. By breaking a number into its prime factors, adjusting the powers, and combining the results, you can calculate the total number of factors quickly and accurately. This method provides clarity, speed, and certainty, while the real insight comes from understanding why we add one to the powers and how that leads to the total number of combinations possible.
Counting the number of factors of a number may appear simple for small numbers, but it becomes tedious when the number grows larger. Imagine trying to list all the factors of 240 by hand. You may get some correct, but it is easy to miss or duplicate others. The smarter approach is to rely on prime factorization, which not only saves time but also ensures accuracy.
The process begins by breaking the number into its prime factors. For example, 240 can be expressed as 2⁴ × 3¹ × 5¹. Factorizing carefully is crucial. Always make sure the bases are prime, and combine powers if the same base appears more than once.
Once you have the prime factorization, add 1 to each exponent. This step reflects the number of ways each prime can contribute to forming factors, including the possibility of not being chosen at all. In the case of 240, adding one to each power gives 5, 2, and 2.
Multiply these values to find the total number of factors. For 240 or 2⁴ × 3¹ × 5¹, that becomes 5 × 2 × 2 = 20. Thus, 240 has exactly 20 factors.
The logic comes from combinations. For four 2s, you can choose zero, one, two, three, or four of them. That is five choices. For a single 3, you can either choose it or not, giving two choices. The same applies for the single 5. Multiplying these options shows the total number of unique factors.
One frequent error is failing to combine the powers of the same prime base. For example, in 1500, if you leave 5 and 5² separate, you will calculate incorrectly. Always combine them into 5³ before proceeding to the next steps.
The method of finding factors through prime factorization is more than a shortcut; it is a lesson in clarity. What first appears long and uncertain becomes simple when approached with structure. The GMAT rewards not brute force but disciplined reasoning and calm under pressure. By practicing this approach, you train your mind to seek patterns rather than chaos, certainty rather than guesswork. To truly master it, exposure under real test-like conditions is essential. A comprehensive GMAT test series with a high number of mocks allows you to apply this method repeatedly, under time limits, until it feels effortless.
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