The question present around at 16 mins (least positive integer where the sum of factors is 24), cant we have Case 3 with factors 2*3*4 where the least positive integer is coming as 360?

Question

Simran Mehta · Accepted Answer

Yes, you are correct. We can have the case with factors 2&times;3&times;4
Understanding the Relationship Between Prime Factorization and Number of Factors:If n has a prime factorization of the form n = p1^e1 * p2^e2 * ... * pk^ek, where p1, p2, ..., pk are distinct primes and e1, e2, ..., ek are their respective powers, then the total number of factors of n is given by:  (e1 + 1)(e2 + 1) ... (ek + 1)
Identify the Factor Combinations That Multiply to 24:- The number 24 can be expressed as a product of integers in several ways:  24 = 24 * 1  24 = 12 * 2  24 = 8 * 3  24 = 6 * 4  24 = 6 * 2 * 2  24 = 4 * 3 * 2
Choosing the Smallest Combination:To minimize n, we want to minimize the exponents and use the smallest primes possible.
Trying Different Combinations:- For 24 = 24 * 1:  - n = p1^23  - n = 2^23 (since 2 is the smallest prime)  - This gives a very large number.- For 24 = 12 * 2:  - n = p1^11 * p2^1  - n = 2^11 * 3^1 = 2048 * 3 = 6144- For 24 = 8 * 3:  - n = p1^7 * p2^2  - n = 2^7 * 3^2 = 128 * 9 = 1152- For 24 = 6 * 4:  - n = p1^5 * p2^3  - n = 2^5 * 3^3 = 32 * 27 = 864- For 24 = 6 * 2 * 2:  - n = p1^5 * p2^1 * p3^1  - n = 2^5 * 3^1 * 5^1 = 32 * 3 * 5 = 480- For 24 = 4 * 3 * 2:  - n = p1^3 * p2^2 * p3^1  - n = 2^3 * 3^2 * 5^1 = 8 * 9 * 5 = 360
Identifying the Smallest Value:- By comparing the values from different combinations, the smallest positive integer with exactly 24 factors is 360.

The question present around at 16 mins (least positive integer where the sum of factors is 24), can't we have Case 3 with factors 2*3*4 where the least positive integer is coming as 360?

The question present around at 16 mins (least positive integer where the sum of factors is 24), can't we have Case 3 with factors 234 where the least positive integer is coming as 360?