Problema Solution
The members of the flag squad wanted to arrange themselves into rows with the same number of squad members in each row. They tried rows of 2, 3, and 4, but there was aways one squad member missing. Finally, they were able to arrnage themselves into exactly five in each row, What is the lesst number of members in the flag squad?
I found the answer 25 by using trial and error but want to find out a way to solve this using least common multiple or greatest common factor.
Answer provided by our tutors
Since the members can be arranged into exactly 5 in each row we can assume that the number of members is
5x, where x > 0, x is integer, x = number of rows
they tried rows of 2, 3, and 4, but there was always 1 squad member missing that is
if we divide 5x with the LCM(2,3,4) = 12 we will get a reminder of -1 (LCM = Least Common Multiple)
5x = 12*q - 1, q is the quotient, -1 is the reminder (since 1 squad member is missing)
for q = 1 we have 5x = 11 and 11 is not dividable by 5 thus q<>1
for q = 2 we have 5x = 23 and 23 is not dividable by 5 thus q<>2
for q = 3 we have 5x = 35 and 35 id dividable by 5 thus the least number of members is 35
indeed
35 = 2*13 - 1
35 = 3*12 - 1
35 = 4*9 - 1
and 35 members can be arranged in 7 rows with exactly 5 members in each row.
Lets consider the answer you found
25 = 2*13 - 1 the condition for the members being arranged in 2 rows and only 1 member is missing is satisfied
25 = 3*9 - 2 thus the second condition for the members being arranged in 3 rows and only 1 member is missing is not satisfied thus 25 is not a solution.