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 positive 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 positive factors.
What is the number of factors of 4N (where N is a natural number)?
!)2N is having 20 factors.
2)3N is having 15 factors
Statement 1:
Case 1: 2N = 2x�, where N = x� and x≠2
Since 2N = 2¹x�, the number of factors for 2N = (1+1)(9+1) = 20.
Case 2: 2N = 2x�y, where
N = x�y, x≠2, y≠2 and x≠y.
Since 2N = 2¹x�y¹, the number of factors for 2N = (1+1)(4+1)(1+1) = 20.
Since each option for N will yield a different number of factors for 4N, INSUFFICIENT.
Other cases:
Case 3: 2N = 2(2²x�) = 2³x�, where N = 2²x� and x≠2
Since 2N = 2³x�, the number of factors for 2N = (3+1)(4+1) = 20.
Case 4: 2N = 2(2³x³) = 2�x³, where N = 2³x³ and x≠2
Since 2N = 2�x³, the number of factors for 2N = (4+1)(3+1) = 20.
Case 5: 2N = 2(2³xy) = 2�xy, where N = 2³xy and x≠2, x≠2, y≠2 and x≠y
Since 2N = 2�x¹y¹, the number of factors for 2N = (4+1)(1+1)(1+1) = 20.
Case 6: 2N = 2(2�x) = 2�x, where N = 2�x and x≠2
Since 2N = 2�x¹, the number of factors for 2N = (9+1)(1+1) = 20.
Case 7: 2N = 2(2¹�) = 2¹�, where N = 2¹�
Since 2N = 2¹�, the number of factors for 2N = 19+1 = 20.
Statement 2:
Only three cases are possible:
Case 1: 3N = 3(3x�) = 3²x�, where
N = 3x� and x≠3
Since 3N = 3²x�, the number of factors for 3N = (2+1)(4+1) = 15.
Case 2: 3N = 3(3³x²) = 3�x², where N = 3³x² and x≠3
Since 3N = 3�x², the number of factors for 3N = (4+1)(2+1) = 15.
Case 3: 3N = 3(3¹³) = 3¹�, where N = 3¹³.
Since 3N = 3¹�, the number of factors for 3N = 14+1 = 15.
Since each option for N will yield a different number of factors for 4N, INSUFFICIENT.
Statements combined:
Only the value in blue (N = x�y = 3x�) satisfies both statements.
Since N=3x�, 4N = 4(3x�) = 2²3¹x�, with the result that the number of factors for 4N = (2+1)(1+1)(4+1) = 30.
The correct answer is
C.
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