How many positive integers less than 500 have exactly 15 positive integer factors. Answer
Answers
Answer:
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,72,74,76,78,80,82,84,86,88,90,92,94,96,98,
100 ...etc
I hope it's helpful for you
Answer:
The correct answer is, there are only 3 such positive integers less than 500, which have exactly 15 positive positive integer factor such as 144, 324, 400
Step-by-step explanation:
Total number of divisors D of any number N with prime factorisation a^p × b^q × c^r is given by formula D = (p+1)(q+1)(r+1)
We need to find numbers with 15 divisors
15 can be split into factors only as 1×15 & 3×5
So, the number having 15 divisors must have it's prime factorisation either as a^14 OR as a^2 × b^4
To find number a^14 less than 500 is impossible case as even 2^9 attains value 512 which is more than 500
So we have to search numbers of type a^2 × b^4 = (a×b^2)^2
As 22^2 < 500 < 23^2, we must choose a & b such that ab^2 < 23
We observe that 5^2 = 25 > 23 , So we can choose primes a & b = 2,3,5 but < 7 so that ab^2 < 23 i.e. (2,3), (3,2), (5,2) are the only prime pairs which can give us our required numbers !
a = 2 & b = 3 gives us N = ( 2× 3^2 )^2 = 324
a = 3 & b = 2 gives us N = ( 3× 2^2 )^2 = 144
a = 5 & b = 2 gives us N = ( 5×^2 )^2 = 400
So there are only 3 numbers!