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02 Jul 2025, 09:07
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The correct answer choice is (D) 4
To solve this question, one must apply the rules of prime factorization to the number of positive factors, 24. To find the total number of positive factors of a number, one should 1) complete the prime factorization of the number 2) add 1 to each exponent 3) multiply together to find the total number of positive factors.
Since we only have "n" for the number, we cannot simply prime factorize and get the answer. Instead we work backwards from the total number of prime factors (24) until we get the prime factorization of "n".
I took 24 and broke it down into the smallest possible prime numbers (basically a prime factorized it) to get:
2 x 2 x 2 x 3 = 24
Each of those could represent step 2 of the steps for finding the total number of factors. To restate that, each of the 2's and the 3 could represent the exponent +1 that then all got multiplied together to get to 24.
(1+1)(1+1)(1+1)(2+1) = 24
Since the greatest possible breakdown results in 4 factors, there are 4 possible distinct prime factors of "n".
To solve this question, one must apply the rules of prime factorization to the number of positive factors, 24. To find the total number of positive factors of a number, one should 1) complete the prime factorization of the number 2) add 1 to each exponent 3) multiply together to find the total number of positive factors.
Since we only have "n" for the number, we cannot simply prime factorize and get the answer. Instead we work backwards from the total number of prime factors (24) until we get the prime factorization of "n".
I took 24 and broke it down into the smallest possible prime numbers (basically a prime factorized it) to get:
2 x 2 x 2 x 3 = 24
Each of those could represent step 2 of the steps for finding the total number of factors. To restate that, each of the 2's and the 3 could represent the exponent +1 that then all got multiplied together to get to 24.
(1+1)(1+1)(1+1)(2+1) = 24
Since the greatest possible breakdown results in 4 factors, there are 4 possible distinct prime factors of "n".
02 Jul 2025, 09:13
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we have to find the range,
1. Minimum number of distinct prime factors (k_min ):
This occurs when 24 is expressed as a single factor (or equivalently, one prime factor with a large exponent).
a1 +1=24
a1 =23
So, k_min =1.
2. Maximum number of distinct prime factors (k_max):
This occurs when 24 is factored into the maximum possible number of integers, all of which are greator than or equal to 2. To maximize the number of factors, we should use the smallest possible values for each factor, which is 2.
Let's express 24 as a product of 2s and possibly one other factor:
24=2*12 (2 factors)
24=2*2*6 (3 factors)
24=2*2*2*3 (4 factors)
The next factor would have to be 2, making the product 16*2=32. Since 32 >24, we cannot have 5 or more factors (each > or = 2).
So, the maximum number of factors in the product (a1 +1)(a2 +1)⋯(ak +1) is 4.
This corresponds to a1 +1=2,a2 +1=2,a3 +1=2,a4 +1=3.
This means a1 =1,a2 =1,a3 =1,a4 =2.
In this case, k=4. (e.g., n=2^1x3^1x5^1x7^2)
So, k_max =4.
Therefore, Range = k_max −k_min =4−1=3.
1. Minimum number of distinct prime factors (k_min ):
This occurs when 24 is expressed as a single factor (or equivalently, one prime factor with a large exponent).
a1 +1=24
a1 =23
So, k_min =1.
2. Maximum number of distinct prime factors (k_max):
This occurs when 24 is factored into the maximum possible number of integers, all of which are greator than or equal to 2. To maximize the number of factors, we should use the smallest possible values for each factor, which is 2.
Let's express 24 as a product of 2s and possibly one other factor:
24=2*12 (2 factors)
24=2*2*6 (3 factors)
24=2*2*2*3 (4 factors)
The next factor would have to be 2, making the product 16*2=32. Since 32 >24, we cannot have 5 or more factors (each > or = 2).
So, the maximum number of factors in the product (a1 +1)(a2 +1)⋯(ak +1) is 4.
This corresponds to a1 +1=2,a2 +1=2,a3 +1=2,a4 +1=3.
This means a1 =1,a2 =1,a3 =1,a4 =2.
In this case, k=4. (e.g., n=2^1x3^1x5^1x7^2)
So, k_max =4.
Therefore, Range = k_max −k_min =4−1=3.
