Question: Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?Unique factors of a number can be calculated by finding all prime factors, adding 1 to each unique prime factor quantity, then multiplying the resulting numbers.
The number 12 can be broken down to (2)(3)(2), this break down means we need to find a value for N with two prime factors with a quantity of one each, and one prime factor with a quantity of two.
Any three prime factors will do.
For example...
\(\mathtt{(3^1)(2^2)(7^1) = 84} \to \mathtt{12 unique factors\)
OR
\(\mathtt{(7^1)(5^2)(11^1) = 1925} \to \mathtt{12 unique factors\)
OR
\(\mathtt{(5^1)(3^2)(2^1) = 90} \to \mathtt{12 unique factors\)
The following technique is used to determine prime factors and factor number totals...
\[\text{Prime Factorization Calculation }\\
\text{Example: N = 90} \\
\begin{pmatrix*}
&&90&& \\
&\swarrow&|&\searrow&\\
2&&\downarrow&&5\\
&&9&&\\
&\swarrow&&\searrow&\\
3&&&&3
\end{pmatrix*}
\rightarrow \text{Prime Factors if N are } 2,3,3,5\\
\]
\[\text{Unique Factor Calculation }\\
\text{Example: N = 90}\\
\begin{pmatrix*}
Prime\text{ Factor } &\text{ Add 1 }& \text{ # of Factors } \\
2^1 & 2^{1+1} & 2 \\
3^2 & 3^{2+1} & 3 \\
5^1 & 5^{1+1} & 2\\
\end{pmatrix*}
\rightarrow\text{ Total number of factors are (2)(3)(2) = 12}\\
\\
\text{Prime Factors of N are } 2,3,3,5\\
\text{Unique factors remove duplicates, leaving } 2,3,5 = \text{ 3 Total Unique Factors} \]
Correct Answer: B. 3 _________________
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