Let's find the total number of integers in the range $[100, 700]$. The total number of integers is $700 - 100 + 1 = 601$. Thus, the total number of integers is $601$.

Let $n(3)$ be the number of multiples of 3 in the given range. The first multiple of 3 is $102 = 3 \times 34$ and the last multiple of 3 is $699 = 3 \times 233$. So, $n(3) = 233 - 34 + 1 = 200$.

Let $n(4)$ be the number of multiples of 4 in the given range. The first multiple of 4 is $100 = 4 \times 25$ and the last multiple of 4 is $700 = 4 \times 175$. So, $n(4) = 175 - 25 + 1 = 151$.

Let $n(12)$ be the number of multiples of 12 in the given range. The first multiple of 12 is $108 = 12 \times 9$ and the last multiple of 12 is $696 = 12 \times 58$. So, $n(12) = 58 - 9 + 1 = 50$.

The number of elements in A is given by the total number of elements minus (multiples of 3 + multiples of 4) + (multiples of 12)
$n(A) = 601 - (200 + 151) + 50 = 601 - 351 + 50 = 250 + 50 = 300$.

Therefore, the number of elements in A is $300$.