Question

Shannon is making identical balloon arrangements for a party. She has 32 maroon balloons, 24 white balloons, and 16 orange balloons. She wants each arrangement to have the same number of each color. What is the greatest number of arrangements she can make if every balloon is used?

Answer 1

Hint: Shannon has 32 maroon balloons, 24 white balloons, and 16 orange balloons and there is a condition that each arrangement has the same number of each color so the possible number of arrangements is calculated by finding the H.C.F (highest common factor) of these three numbers (32, 24 and 16). Now, H.C.F of any three numbers is calculated by writing the prime factorization first, and then we will find the factors which are common in all the three prime factorizations. That common factor is the H.C.F of the three numbers.

Complete step by step answer:
In the above problem, Shannon has to arrange 32 maroon balloons, 24 white balloons, and 16 orange balloons in such a manner so that each arrangement has the same number of each color which we are going to find by writing the H.C.F(highest common factor) of three numbers (32, 24 and 16).
We know that the H.C.F of three numbers is calculated by writing the prime factorizations of the three numbers and then the factor which is common among the three prime factorizations is the H.C.F.
Writing the prime factorization of 32, 24 and 16 we get:
$\begin{align}
  & 32=2\times 2\times 2\times 2\times 2 \\
 & 24=2\times 2\times 2\times 3 \\
 & 16=2\times 2\times 2\times 2 \\
\end{align}$
Now, we are underlining the common factors among the three numbers (32, 24 and 16) and we get,
$\begin{align}
  & 32=\underline{2\times 2\times 2}\times 2\times 2 \\
 & 24=\underline{2\times 2\times 2}\times 3 \\
 & 16=\underline{2\times 2\times 2}\times 2 \\
\end{align}$
From the above, you can see that the common factors which we are getting from the above are as follows:
$2\times 2\times 2$
Multiplying the 3 factors will give us:
$=8$
Hence, the number of arrangements that are possible is 8.

Note: You might be thinking that how we know when to use H.C.F. The answer lies in the two statements given in the above problem, the condition for the arrangements is that each of the arrangements contains the same number of each color and are asked to find the greatest arrangements. So, from these conditions, we need to find the H.C.F of the three numbers (32, 16, and 24) and H.C.F is the highest common factor (means maximum common factors in the three numbers).