M4maths Previous Puzzles

11 Jul 2018 Perfect Square

How many factors of 16295655600 are perfect squares?

OPtion

1) 12
2) 8
3) 16
4) 18
5) 25
6) 24
7) 30
8) 40
9) 15
10) None of these

Solution

16295655600 = 53^1 * 83^1 * 7^3 * 5^2 * 2^4 * 3^3

Any factor of this number should be of the form 53^a * 83^b * 7^c * 5^d * 2^e * 3^f.

For the factor to be a perfect square a,b,c,d,e,f have to be even.

a can take values 0,
b can take values 0,
c can take values 0, 2
d can take values 0, 2
e can take values 0, 2, 4
and f can take values 0, 2


Total number of perfect squares = 1*1*2*2*3*2 = 24

Option 6)

Correct Option: 6 Best Solution (3)

raksha s shetty   6 years AGO

16295655600 factors are 1*(2^4)*(3^3)*(7^3)*(5^2)*4399
Let 16295655600=(2^a)*(3^b)*(5^c)*(7^d)*(4399^e)
For factors to be perfect squares powers of a,b,c,d,e should be even
Therefore a =can be 0,2,4
b=can be 0,2
c=can be 0,2
d=0,2
e=0
Therefore 3*2*2*2=24 factors are perfect squares

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Varun   6 years AGO

16295655600 = 2^4 * 3^2 * 5^2 * 7^3 * 53 * 83
So to become perfect square factor, number should be repeated even number of times
And in 2,3, 5, 7
They are repeated 3, 2, 2 , 2 times
so total = 3*2*2*2 = 24
= Option (6)

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SAYAN CHATTERJEE   6 years AGO

16295655600=
5^2*2^4*3^3*7^3*4399
Hence, no.of square factors is (2*3*2*2)= 24.
Option 6.

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