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4 x 3 x 49 = 588

Factors (from a website):

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588,
-1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42, -49, -84, -98, -147, -196, -294, -588     Pick the ones you like and count them up !

Here's  a way to figure this

2^1 = 2

2^2 = 4

2 * 3  =  6

2^2 * 3  = 12

2* 7   =  14

2^2 * 7 = 28

2 * 3 * 7  = 42

2^2 *3 * 7  = 84

2*7^2 =  98

2^2 * 7^2  = 196

2 *3 * 7^2  =  294

2^2 * 3 * 7^2 =  588

[ Note....negative factors  are not "natural"  numbers ]

How many even natural-number factors does \(n = (2^2)(3^1)(7^2)\) have?

\(\begin{array}{|llcll|} \hline 2^{\color{red}2} * & 3^{\color{blue}1} * 7^{\color{green}2} \\\\ ({\color{red}2}+1 ) *& ({\color{blue}1}+1) * ({\color{green}2}+1) = 3*2*3 &=& 18~ \text{divider} \\ & ({\color{blue}1}+1) * ({\color{green}2}+1) = 2*3 &=& 6~ \text{odd divider} \qquad (\text{odd*odd=odd})\\ \hline &&& 18-6 = \mathbf{12}~ \text{even divider} \\ \hline \end{array} \)

Thank you all so much for your answers, they helped me a lot!! :))