# Three lines

At 6 am, three bus lines are departing from the station. The first line has an interval of 24 minutes. The second line has an interval of 15 minutes. The third line runs at regular intervals of more than 1 minute. The third line runs at the same time as the first, also at the same time as the second, and sometimes runs alone. Find the interval of the 3rd line.

Result

t3 =  3 min

#### Solution:

$t_{ 1 } = 24 \ min \ \\ t_{ 2 } = 15 \ min \ \\ \ \\ t_{ 3 }>1 \ \\ \ \\ SD(t_{ 1 },t_{ 2 }): 1, 3 \ \\ \ \\ 24 = 2^3 \cdot 3 \\ 15 = 3 \cdot 5 \\ \text{GCD}(24, 15) = 3 = 3\\ \ \\ \ \\ t_{ 3 } = GCD(t_{ 1 },t_{ 2 }) = GCD(24,15) = 3 = 3 \ \text { min }$

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