Question

A farmer has 2494 sheep and 2193 lambs. He farms them into flocks, keeping sheep and lambs separate and having the same number of animals in each flock. If these flocks are as large as possible. Find the total number of flocks required for the purpose.

Answer 1

Hint: It is given to us that the number of animals in each flock is the same and sheep and lambs are separate. This means there is no sheep in the lamb's flock and vice-versa. To find the number of animals, we need to find the highest common factor of total number of sheep and number of lambs. Then we need to add the number of flocks formed by sheep and lambs to get the total number of flocks.

Complete step-by-step answer:
First of all, we will find the prime factors of 2494.
$\begin{align}
  & \ \ 2\left| \!{\underline {\,
  2494 \,}} \right. \\
 & 29\left| \!{\underline {\,
  1247 \,}} \right. \\
 & 43\left| \!{\underline {\,
  43 \,}} \right. \\
 & \ \ \ \ \ 1
\end{align}$
Thus, factorisation of 2494 is 2 x 29 x 43.
Now, we will find the prime factors of 2193
\[\begin{align}
  & \ \ 3\left| \!{\underline {\,
  2193 \,}} \right. \\
 & 17\left| \!{\underline {\,
  731 \,}} \right. \\
 & 43\left| \!{\underline {\,
  43 \,}} \right. \\
 & \ \ \ \ \ 1
\end{align}\]
Thus, factorisation of 2193 is 3 x 17 x 43.
The only common factor of 2494 and 2193 is 43. Therefore, 43 is the highest common factor of 2494 and 2193.
This means, each flock will have 43 animals each.
Now, to find the total number of flocks, we will find the number of sheep flocks and number of lamb flocks and add them.
To get the number of sheep flocks, we will divide 2494 by 43.
$\dfrac{2494}{43}=58$
And to get the number of lamb flocks, we will divide 2193 by 43.
$\dfrac{2193}{43}=51$
Thus, the number of sheep flocks will be 58 and the number of lamb flocks will be 51.
Total number of flocks will be 58 + 51 = 109.

Note: To be quick in finding the factors of a number, students are advised to be well-versed in the multiplication table of prime numbers as higher tables can be a little tricky. Also, go through the question thoroughly to understand what actually has to be found and what are the conditions that govern the solution.