Q. How many numbers. are there within 500 which has exactly 24 factors.

• If you factor a number into its prime factors:
  
  N = p1^e1 * p2^e2 * ... * pn^en
  
  Then the number of factors is (e1 + 1)(e2 + 1) ... (en + 1)
  
  CASE 1: N has 1 factor
  N = p1^e1
  number of factors = e1 + 1 = 24
  e1 = 23
  The smallest such number is 2^23 which is bigger than 500
  
  CASE 2: N has 2 factors
  N = p1^e1 * p2^e2
  number of factors = (e1 + 1)(e2 + 1) = 24
  
  Ways to factor 24 into 2 factors bigger than 1:
  2*12, 3*8, 4*6
  
  Solutions for (e1,e2)
  (1,11), (2,7), (3,5)
  
  None of these give an N less than 500.
  2^11 * 3^1 = 6144
  2^7 * 3^2 = 1152
  2^5 * 3^3 = 864
  
  CASE 3: N has 3 factors
  N = p1^e1 * p2^e2 * p3^e3
  number of factors = (e1 + 1)(e2 + 1)(e3 + 1) = 24
  
  Ways to factor 24 into 3 factors bigger than 1:
  2*2*6, 2*3*4
  
  Solutions for (e1,e2,e3)
  (1,1,5), (1,2,3)
  
  N = 480 = 2^5 3^1 5^1
  N = 360 = 2^3 3^2 5^1
  
  
  CASE 4: N has 4 factors
  N = p1^e1 * p2^e2 * p3^e3 * p4^e4
  number of factors = (e1 + 1)(e2 + 1)(e3 + 1)(e4 + 1) = 24
  
  Ways to factor 24 into 3 factors bigger than 1:
  2*2*2*3
  
  Solutions for (e1,e2,e3,e4)
  (1,1,1,2)
  
  N = 420 = 2^2 3^1 5^1 7^1
  
  
  So the numbers less than 500 with 24 factors are
  
  N = 360 = 2^3 3^2 5^1
  N = 420 = 2^2 3^1 5^1 7^1
  N = 480 = 2^5 3^1 5^1
• 10 years agoHelpfull: Yes(47) No(1)
• number of factors can be found by : P^a*Q^b*R^c=(a+1)(b+1)(c+1)
  24 can be written as=3*2*2*2, 6*2*2, 4*3*2
  2^2*3*5*7=420
  2^5*3*7=480
  2^3*3^2*5=360
  
  Ans: 3 numbers under 500 has 24 factors
• 10 years agoHelpfull: Yes(14) No(2)
• question was how many numbers have more than 24 factors....its not exactly 24.
  options:
  1>1
  2>2
  3>3
  4> none of them
  ans-none of them
• 10 years agoHelpfull: Yes(5) No(5)
• P^a*Q^b*R^c=(a+1)(b+1)(c+1)
  24 can also be written as=3*2*2*2, 6*2*2, 4*3*2...
  2^2*3*5*7=420
  2^5*3*5=480
  2^3*3^2*5=360
  
  so there r 3 nums
• 10 years agoHelpfull: Yes(3) No(0)
• you can calculate factors of any given no as
  take no is 40 then we need to factories into 2*2*2*5 that will give us like
  (2^0+2^1+2^2+2^3)*(5^0=5^1)=4*2=8 factors of 40.
  so now we need to calculate factors with 24
  after cheking answers will be three because
  a 360
  b 420
  c 480
  have 24 factors.
• 10 years agoHelpfull: Yes(1) No(0)
• options are
  A.1
  B.2
  C.3
  D. Cant determine
• 10 years agoHelpfull: Yes(0) No(0)
• 360,420,480
• 10 years agoHelpfull: Yes(0) No(0)
• 420, 360, 480
• 10 years agoHelpfull: Yes(0) No(0)