Six bells ring at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. Once, when they start ringing simultaneously for the first time, then determine how many times they will ring together in a continuous span of 30 minutes?

Six bells ring at intervals of 2, 4, 6, 8, 10, and 12 seconds respectively. They all start ringing together at time $t = 0$.

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To find when all bells ring together, the time must be a common multiple of all their intervals. This requires finding the LCM (Least Common Multiple) of 2, 4, 6, 8, 10, and 12.

Prime factorization:

$2 = 2$

$4 = 2^2$

$6 = 2 \times 3$

$8 = 2^3$

$10 = 2 \times 5$

$12 = 2^2 \times 3$

Taking the highest power of each prime factor:

$\text{LCM} = 2^3 \times 3 \times 5$

$\text{LCM} = 8 \times 3 \times 5$

$\text{LCM} = 120$ seconds

All 6 bells ring together every 120 seconds.

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Converting 30 minutes to seconds:

$30 \times 60 = 1800$ seconds

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The bells ring together at $t = 0, 120, 240, 360, ...$ up to 1800 seconds.

Number of intervals:

$\dfrac{1800}{120} = 15$ intervals

Since there is a ringing at the start ($t = 0$) plus one ringing after each of the 15 intervals:

Number of times $= 15 + 1 = 16$

Therefore, the bells will ring together 16 times in 30 minutes.