eshwarjayanth wrote:X=12Y, where Y is a prime number greater than 3. How many different positive divisors does X^2 having including X.
7
12
25
35
45
To count 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 the reasoning. To determine how many factors can be formed from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor:
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, we get 4*3 = 12 possible factors.
To verify, here are the 12 positive factors of 72:
1 72
2 36
3 24
4 18
6 12
8 9
Onto the problem above:
Plug in y=5.
Then x = 12 * 5 = 2² * 3 * 5
x² = (2² * 3 * 5)² = 2� * 3² * 5².
Adding 1 to each exponent and multiplying, we get:
Number of factors = (4+1)(2+1)(2+1) = 5*3*3 = 45.
The correct answer is
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