Question

When teams of the same sizes are formed from three groups of 512, 430, and 489 students separately 8, 10, and 9 students respectively are left out. What could be the largest size of the team?
(A) 6
(B) 12
(C) 18
(D) 20

Answer 1

Hint: We start solving this problem by first subtracting the left-out students from the given three groups. Then we acknowledge that the size of the team is a factor of the size of groups formed after subtracting the left-out students. As we need to find the largest size of the team, we should find the HCF of the numbers obtained. Then we write the factorization of those three numbers and find the factors common to them and multiply them to find the HCF, which is the largest size of the team.

Complete step-by-step solution
We are given that there are three groups of 512, 430, and 489 students.
We are also given that when teams of the same size are made from these groups separately then 8, 10, and 9 students are left out.
So, if 8 students are left out from the group of 512 students, then the number of students in that group that are in the team are,
$\begin{align}
  & \Rightarrow 512-8 \\
 & \Rightarrow 504 \\
\end{align}$
Similarly, if 10 students are left out from the group of 430 students, then the number of students in that group that are in the team are,
$\begin{align}
  & \Rightarrow 430-10 \\
 & \Rightarrow 420 \\
\end{align}$
if 9 students are left out from the group of 489 students, then the number of students in that group that are in the team are,
$\begin{align}
  & \Rightarrow 489-9 \\
 & \Rightarrow 480 \\
\end{align}$
So, if we remove the left-out students from the three groups, then the number of students in the three groups are 504, 420 and 480.
Now we are given that groups of the same size are formed from them. So, let us assume that the size of those groups is $x$.
 So, as $x$ is the size of groups made from three groups 504, 420, and 480, then $x$ is a factor of 504, 420, and 480.
We need to find the largest size of the team, that is we need to find the greatest common factor of 504, 420, and 480.
So, let us find the HCF of 504, 420, and 480.
First let us factorise the above numbers. Then we get,
\[\begin{align}
  & \Rightarrow 504=2\times 2\times 2\times 3\times 3\times 7 \\
 & \Rightarrow 420=2\times 2\times 3\times 5\times 7 \\
 & \Rightarrow 480=2\times 2\times 2\times 2\times 2\times 3\times 5 \\
\end{align}\]
From above we can see that all of them have two 2’s and one 3.
So, we get the HCF of 504, 420 and 480 as,
$\begin{align}
  & \Rightarrow HCF=2\times 2\times 3 \\
 & \Rightarrow HCF=12 \\
\end{align}$
So, we get the HCF as 12.
So, the largest size of the team is 12.
Hence the answer is Option B.

Note: The common mistake one does in this question is, that does not subtract the left-out students from the given groups and find the HCF of those numbers to find the largest team size. But first, we need to subtract as there are left out students if the teams of the same size are formed.