Q. How many nos. are there within 500 which has exactly 24 factors.
a) 2
b) 3
c) 1
d) 0

• b) 3
  
  no. of factors of p^a*q^b*r^c*...=(a+1)*(b+1)*(c+1)*...
  
  24 = 3*2*2*2 = 4*3*2 = 6*2*2
  
  2^2*3*5*7 = 420
  2^3*3^2*5 = 360
  2^5*3*5 = 480
  
  3 numbers within 500 has exactly 24 factors.
• 10 years agoHelpfull: Yes(48) No(7)
• 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(7) No(1)
• 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(6) No(0)
• prime factors of 24 are 2,2,2,3 that means (1+1)(1+1)(1+1)(2+1) as no of factors are (x+1)(y+1)(z+1)....
  where x,y,z are the powers of individual prime factors so from above take 1,1,1,2 as powers and check by putting prime nos so the only case found was 3*5*7*2^2= 420
• 10 years agoHelpfull: Yes(3) No(7)
• hai rakesh, how become 2^2*3*5*7=420, then what is exactly 24 factors
• 10 years agoHelpfull: Yes(2) No(1)
• hey lalit,how did yod find 420?? plz explain
• 10 years agoHelpfull: Yes(1) No(5)
• @Rakesh
  how u got 420.....what is the basic idea..how do u came to the L.H.S part 2^2*3*5*7 ..if u will ellaborate it wil be easy to understand
• 10 years agoHelpfull: Yes(1) No(0)
• @ RAKESH : not getting please explain
• 10 years agoHelpfull: Yes(1) No(0)
• 240 also have exactly 24 factors
  2^2*3*4*5
  same way 384 also have 24 factors
  2^5*3*4
  and 420,360,480
  total no of numbers within 500 which has 24 factors is 5
  
• 10 years agoHelpfull: Yes(1) No(1)
• 2^2*3*5*7=420
  no of factors = (2+1)(1+1)(1+1)(1+1)= 24
  option c
• 10 years agoHelpfull: Yes(0) No(7)
• its 420 and 480 also 420=(2^2*3*5*7) and 480=(2^5*3*5)
• 10 years agoHelpfull: Yes(0) No(3)
• 4
  A-B,B-C,C-B,B-A,A-B-C OR A-C-B.....A-C,C-B,B-C,C-A,A-B-C OR A-C-B
• 10 years agoHelpfull: Yes(0) No(0)