A SCHOOL IS DIVIDED INTO 4 HOUSES P,Q,R,S. ON SPORTS DAY,STUDENTS LINE UP SO THAT A STUDENT FROM P IS FIRST IN LINE,THEN A STUDENT FROM Q THEN A STUDENT FROM R THEN A STUDENT FROM S THEN A STUDENT FROM P AND SO ON WITH SAME ORDER REPEATING .FROM WHAT HOUSE WILL THE 66TH STUDENT COME? PLEAE EXPLAIN HOW OU WORK IT OUT,THANKS

House P has students numbers

1 5 9 13 17 ........ (4n-3)
House Q has students numbers
2 6 10 14 ...... (4n-2)
etc

4n-3 = 66
4n = 69 --> no whole number solution

4n-2 = 66
4n = 68
n = 17 ----- Yeah
The 66th student is in house Q

or

make 4 columns
P...Q...R....S
1 .. 2 .. 2 .. 4
5 .. 6 .. 7 .. 8
9 .. 10 11 12
13 14 15 16
17 18 19 20
...
65 66 67 68

for P -- remainder of 1 when divided by 4
for Q -- remainder of 2 when divided by 4 ---- 66
for R -- remainder of 3 when divided by 4

To solve this problem, we need to find the pattern of how the students from different houses are lined up.

Let's analyze the pattern:
1st student: House P
2nd student: House Q
3rd student: House R
4th student: House S
5th student: House P
6th student: House Q

From the pattern, we can observe that the houses are repeating in a cycle of 4 (P, Q, R, S).

To find out from which house the 66th student will come, we can divide 66 by 4 and find the remainder.

66 divided by 4 equals 16, with a remainder of 2.

This means that in the first complete cycle of 4 houses, the 66th student is not yet reached.

The remainder of 2 indicates that the 66th student will be from the second position of the cycle, which is House Q.

Therefore, the 66th student will come from House Q.

To verify this, we can count the students up to the 66th position:

1st student: House P
2nd student: House Q

So, the 66th student will be from House Q.