Latest Elitmus Aptitude Question SOLUTION: 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.

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

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

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

hai rakesh, how become 2^2*3*5*7=420, then what is exactly 24 factors

hey lalit,how did yod find 420?? plz explain

@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

@ RAKESH : not getting please explain

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

2^2*3*5*7=420
no of factors = (2+1)(1+1)(1+1)(1+1)= 24
option c

its 420 and 480 also 420=(2^2*3*5*7) and 480=(2^5*3*5)

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