The middle school band has 56 members. The high school band has 96 members. The bands are going to march one after the other in a parade. The director wants to arrange the bands into the same number of columns. What is the greatest number of columns in which the two bands can be arranged if each column has the same number of marchers? How many marchers will be in each column?
To find the greatest number of columns in which the two bands can be arranged, we need to find the greatest common divisor (GCD) of 56 and 96. The GCD represents the largest number that divides both 56 and 96 evenly.
To find the GCD, we can use the Euclidean algorithm:
1. Find the remainder when 96 is divided by 56: 96 % 56 = 40.
2. Find the remainder when 56 is divided by 40: 56 % 40 = 16.
3. Find the remainder when 40 is divided by 16: 40 % 16 = 8.
4. Find the remainder when 16 is divided by 8: 16 % 8 = 0.
Since we have reached a remainder of 0, the GCD is the divisor used in this step, which is 8.
Therefore, the greatest number of columns in which the two bands can be arranged is 8.
To find the number of marchers in each column, we divide the total number of marchers in each band by the number of columns:
For the middle school band: 56 members ÷ 8 columns = 7 marchers per column.
For the high school band: 96 members ÷ 8 columns = 12 marchers per column.
So, there will be 7 marchers per column in the middle school band and 12 marchers per column in the high school band.