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MGMAT - Advanced GMAT Quant Question-

by imhimanshu » Tue Aug 16, 2011 7:58 pm
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
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by Frankenstein » Tue Aug 16, 2011 8:57 pm
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.
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by GMATGuruNY » Wed Aug 17, 2011 2:29 am
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.
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by lunarpower » Thu Aug 25, 2011 2:44 am
mitch's solution is good.

once you realize that y = 2 x 3 x 5 = 30 is the minimum value of y, you can also just list the factors of 30 and count them:
1, 30
2, 15
3, 10
5, 6
there you go -- eight factors.
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by Scott@TargetTestPrep » Tue Dec 12, 2017 9:33 am
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

We are given y^4/60 = integer. In other words:

y^4/(2^2 x 3^1 x 5^1) = integer

Since y must have at least one 2, one 3 and one 5 in order for y^4/60 = integer, the minimum value of y must be (2^1 x 3^1 x 5^1), or 30.

Now, to determine the number of distinct factors, we can use the following shortcut:

The total number of factors of a number can be obtained by multiplying the numbers resulting from adding 1 to the exponents in the prime factorization. Thus, the total number of factors of y is:

(1 + 1) x (1 + 1) x (1 + 1) = 2 x 2 x 2 = 8

Alternately, we could list all factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Thus, y has 8 distinct factors.

Answer: C

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