How many natural numbers less than 500 have exactly 24 factors?
Answers
Answer:
First of all, understand how to count number of factors. Here are steps to find out numbers of factors.
Write the number in form of power to prime numbers.
Add 1 to each exponent and multiply all exponents.
Now, we can go up to 8 power for 2, 5 power for 3, 3 power for 5, 3 power for 7 and 1 power to (11, 13, 17, 19, 23) if we are using number alone.
But we have combinations of prime numbers. We are not gonna use them alone. So if two prime numbers are used, then we can have power as (1,11), (2,7), (3,5). But we can’t make any number below 500. Lowest possible number is 864 which is 2^5 * 3^3 having 24 factors.
Let’s try three prime number combinations. We can have (1,1,5), (1,2,3). So we have 2^5 * 3^1 * 5^1 = 480 and 2^3 * 3^2 * 5^1 = 360. Which have exact 24 factors.
For example,
360 has (1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360) and 480 has (1,2,3,4,5,6,8,10,12,15,16,20,24,30,32,40,48,60,80,96,120,160,240,480).
Let’s have four prime numbers combinations. So we have only one combination as (1,1,1,2). This lead us to number 2^2 * 3*5*7 = 420 which has factors as (1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420).
But if we rearrange prime numbers, we will have 2*3^2*5*7 = 630 which is above 500. So all other possibilities will give number higher than 500.
So, According to me, there are three numbers having factors exactly 24 which are 360, 420 and 480.