Latest Elitmus Aptitude Question SOLUTION: 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

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

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

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

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.

options are
A.1
B.2
C.3
D. Cant determine

360,420,480

420, 360, 480