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Answer: 504


Underlined twice: multiples of 4 (i.e. 2 and 4) but not of 3 (red and green)
                          multiples of 6 (i.e. 2 and 3) but not of 12 (red and blue)

Multiples of 4 but not of 3 includes $\frac23$ of the multiples of 4, starting with the first two (4 and 8)
2016$\div$4 = 504 multiples of 4
504$\div$3 = 168 of those are also multiples of 3
504 $-$ 168 = 336 are underlined exactly twice

2016$\div$6 = 336 multiples of 6
336$\div$2 = 168 multiples of 12
Also 168 multiples of 6 which are not multiples of 12

168 + 336 = 504