The least number which has 24 factors will be 280, when the 24 will be split into 2*3*4 and p,q,r will be 1,2,3. powers will be raised to 2 to power 3, 3 to power 2 and 5 to power 1= 280.?

Yes, you are correct. We can have the case with factors 2×3×4

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.