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 its 360. Help?

Question

Sarthak Singla · Accepted Answer

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^ekThen 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 * 124 = 12 * 224 = 8 * 324 = 6 * 424 = 6 * 2 * 224 = 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 * 1N = p1^23This is not minimal as it involves a very large exponent.
Case 2: 12 * 2N = p1^11 * p2Using the smallest primes, 2^11 * 3 = 2048 * 3 = 6144
Case 3: 8 * 3N = p1^7 * p2^2Using the smallest primes, 2^7 * 3^2 = 128 * 9 = 1152
Case 4: 6 * 4N = p1^5 * p2^3Using the smallest primes, 2^5 * 3^3 = 32 * 27 = 864
Case 5: 6 * 2 * 2N = p1^5 * p2 * p3Using the smallest primes, 2^5 * 3 * 5 = 32 * 3 * 5 = 480
Case 6: 4 * 3 * 2N = p1^3 * p2^2 * p3Using 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.

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?