Hi,

60 = 2^2.3.5
y^4 is divisible by 60.
So, y^4 should be of the form 2^4.3^4.5^4.k^4.
So, y = 2.3.5.k
For the number of factors to be minimum, y should be minimum.
So, y = 2*3*5
If N = a^p*b^q*c^r...where a,b,c are distinct prime factors, then number of factors of N i given by
(p+1)*(q+1)*(r+1)...
Number of factors of y is given by (1+1)(1+1)(1+1) = 8.

imhimanshu wrote:If y^4 is divisible by 60, what is the minimum number of distinct factors that y must have
A) 2
B) 6
C) 8
D) 10
E) 12

My take-

As y^4/60 = k where k is an integer. means, y^4 is completely divisible by 60. Therefore y must have all the prime factors as those of number 60.
That is - 60 = 2*2*3*5
Also, as y contains power of 4... therefore y must have the following factors-
2^4,3^4 and 5^ 4.

Now, what to do next? How to find the minimum number of distinct factors of y? Please clarify
Thanks

60 = 2² * 3 * 5.
For y^4 to be divisible by 2² * 3 * 5, y itself must be divisible by 2, 3 and 5.
Thus, the smallest possible value of y = 2*3*5.

To determine the number of positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply

For example:
72 = 2³ * 3².
Adding 1 to each exponent and multiplying, we get (3+1)*(2+1) = 12 factors.

Here's why:
To determine how many factors can be created from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor and to count the number of ways these choices can be combined:

For 2, we can use 2�, 2¹, 2², or 2³, giving us 4 choices.
For 3, we can use 3�, 3¹, or 3², giving us 3 choices.

Multiplying the number of choices we have of each factor, we get 4*3 = 12 possible factors.

Thus, to count the factors of y = 2*3*5, we add 1 to each exponent and multiply:
Number of factors = (1+1)*(1+1)*(1+1) = 8.

The correct answer is C.