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 the reasoning. 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:
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.
Onto the problem above:
[email protected] wrote:If n has 15 positive divisors, inclusive of 1 and n, then which of the following could be the number of divisors of 3n?
I. 20
II. 30
III. 40
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III only
OA
B
To count the total number of divisors of n, we prime-factorize n, add 1 to each exponent, and multiply.
Since n has exactly 15 positive divisors, the following cases are possible:
Case 1: n = a¹�, where a≠3
Here, the number of factors on n = 14+1 = 15.
Since 3n = 3¹a¹�, the number of factors of 3n = (1+1)(14+1) = 30.
Eliminate C, which does not include statement II.
Case 2: n = 3¹�
Here:
The number of factors of n = 14+1 = 15.
Since 3n = 3¹�, the number of factors of 3n = 15+1 = 16.
Case 3: n = a²b�, where a≠3 and b≠3
Here:
The number of factors of n = (2+1)(4+1) = 15.
Since 3n = 3¹a²b�, the number of factors of 3n = (1+1)(2+1)(4+1) = 30.
Same result as in Case 1.
Case 4: n = 3²b�, where b≠3
Here:
The number of factors of n = (2+1)(4+1) = 15.
Since 3n = 3³b�, the number of factors of 3n = (3+1)(4+1) = 20.
Eliminate A and D, which do not include statement I.
Case 5: n = a²3�, where a≠3
Here:
The number of factors of n = (2+1)(4+1) = 15.
Since 3n = a²3�, the number of factors of 3n = (2+1)(5+1) = 18.
Since statement III is not possible, eliminate E.
The correct answer is
B.
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