In the question (if a positive integer has 24 factors , what is least possible value of that integer,) the video says answer is 420 but since i calculated putting all scenario together it's 360. Help?

Yes, your calculation is correct

To determine the least positive integer with exactly 24 factors, we need to understand how the number of factors of a positive integer is determined by its prime factorization.

If a number N has a prime factorization of the form:
N = p1^e1 * p2^e2 * ... * pk^ek
Then the number of factors of N is given by:
(e1 + 1) * (e2 + 1) * ... * (ek + 1)

We need to find the smallest integer N such that the number of factors equals 24.

Step 1: Prime Factorization and Factors Count

To get 24 as the product of (e1 + 1) * (e2 + 1) * ... * (ek + 1), we can have different combinations:
24 = 24 * 1
24 = 12 * 2
24 = 8 * 3
24 = 6 * 4
24 = 6 * 2 * 2
24 = 4 * 3 * 2

We need to select the combination that leads to the smallest possible N.

Step 2: Finding the Smallest N

Let's test the combinations with the smallest prime numbers.

Case 1: 24 = 24 * 1
N = p1^23
This is not minimal as it involves a very large exponent.

Case 2: 12 * 2
N = p1^11 * p2
Using the smallest primes, 2^11 * 3 = 2048 * 3 = 6144

Case 3: 8 * 3
N = p1^7 * p2^2
Using the smallest primes, 2^7 * 3^2 = 128 * 9 = 1152

Case 4: 6 * 4
N = p1^5 * p2^3
Using the smallest primes, 2^5 * 3^3 = 32 * 27 = 864

Case 5: 6 * 2 * 2
N = p1^5 * p2 * p3
Using the smallest primes, 2^5 * 3 * 5 = 32 * 3 * 5 = 480

Case 6: 4 * 3 * 2
N = p1^3 * p2^2 * p3
Using the smallest primes, 2^3 * 3^2 * 5 = 8 * 9 * 5 = 360

Step 3: Conclusion

The smallest value of N that has exactly 24 factors is 360.

Therefore, the least positive integer with exactly 24 factors is 360.