The smallest possible number with 24 factors should be 360 = 2^3 * 3^2 * 5.
Is my understanding incorrect? (practice problems 2 - number system video)?

Question

S Raghav · Accepted Answer

The logic is almost right, but 360 isn&rsquo;t the smallest number with 24 factors. The total number of factors for a number written as p1^a &times; p2^b &times; p3^c &hellip; is (a+1) &times; (b+1) &times; (c+1) &hellip; = 24.
For 24 factors, one pattern is 4 &times; 3 &times; 2 = 24 &rarr; exponents 3, 2, 1. That&rsquo;s what you&rsquo;ve done with 2^3 &times; 3^2 &times; 5^1 = 360. But to get the smallest number, you give the biggest exponent to the smallest prime, the next biggest to the next smallest prime, and so on.
360 works for that pattern, but it&rsquo;s not the absolute smallest number possible for 24 factors. If you check other factor patterns, the actual smallest number is 840.

Mohit C · Answer

The fastest way is to work backwards from the factor count rule: for p1^a &times; p2^b &times; p3^c, the number of factors = (a+1) &times; (b+1) &times; (c+1).
&bull; 24 can be split as 8 &times; 3 &rarr; exponents 7, 2 &rarr; 2^7 &times; 3^2 = 1152 &bull; Or 6 &times; 4 &rarr; exponents 5, 3 &rarr; 2^5 &times; 3^3 = 864 &bull; Or 4 &times; 3 &times; 2 &rarr; exponents 3, 2, 1 &rarr; 2^3 &times; 3^2 &times; 5 = 360
360 is small, but when you test other splits and assign exponents carefully, 840 ends up being the smallest number with exactly 24 factors.

The smallest possible number with 24 factors should be 360 = 2^3 * 3^2 * 5. Is my understanding incorrect? (practice problems 2 - number system video)?