# There are 120 students waiting to go on field trip. The students are numbered 1 to 120, all even numbered students go on bus1, those divisible by 5 go on bus2 and those whose numbers are divisible by 7 go on bus3. How many students did not get in any bus?

May 14, 2016

$41$ students did not get in any bus.

#### Explanation:

There are $120$ students.

On $B u s 1$ even numbered i.e. every second student goes,

hence $\frac{120}{2} = 60$ students go.

Note that every tenth student i.e. in all $12$ students, who could have gone on $B u s 2$ have left on $B u s 1$.

As every fifth student goes in $B u s 2$, number of students that go in bus (less $12$ which have gone in $B u s 1$) are $\frac{120}{5} - 12 = 24 - 12 = 12$

Now those divisible by $7$ go in $B u s 3$, which is $17$ (as $\frac{120}{7} = 17 \frac{1}{7}$), but those with numbers $\left\{14 , 28 , 35 , 42 , 56 , 70 , 84 , 98 , 105 , 112\right\}$ - in all $10$ have already gone in $B u s 1$ or $B u s 2$.

Hence in $B u s 3$ go $17 - 10 = 7$

Students left are $120 - 60 - 12 - 7 = 41$