Bells tolls 18 time a day simultaneously.

Question

In Mahabalipuram temple there are some magical bells which toll $18$ times in a day, simultaneously. But every bells tolls at a different interval of time, but not in a fraction of minutes. The maximum number of bells in the temple?

I got this question in the test today. Here is the solution but I am unable to get it. Can anyone Elaborate it to me.

Ans- Since these bells tolls $18$ times in $24$ hours.

So the minute time interval when they toll together $= 24/18 × 60 = 80$ minutes.

So the required number of bells $=$ Total number of different factors of $80$.

Now since $80 = 2^4 × 5^1$

Total number of factors $= ( 4 + 1 ) ( 1 + 1 ) = 10$

Thus the maximum number of bells $= 10$.

Answer 1

It seems that the each bell rings at regular intervals, with gaps of an integer number of minutes, and they ring simultaneously $18$ times every $24$ hours ($1440$ minutes).

So they ring simultaneously every $\frac{1440}{18}= 80$ minutes.

So for each the gap must divide $80$.

You can use the number-of-divisors function to count the number of possibilities, by looking at the prime factorisation of $80$, but in any case, the ten are $1, 2, 4, 5, 8, 10, 16, 20, 40, 80$, and so the bells must have distinct gaps from this set, making the maximum possible number of bells ten.

Answer 2

first bell rings at $1$ minute interval for $18$ times consuming $1 \times 18$ minutes, second bell rings at $2$ minutes interval for $18$ times consuming $2 \times 18$ minutes, $\dots$.

Thus, total $1 \times 18 + 2 \times 18 + 3 \times 18 + \dots + n \times 18 \le 24 \times 60$.

So, $18(1+2+3+4+\dots+n) \le 1440$.

Integer value of $n = 12$, hence $12$ is the answer.