Problem of the Week Problem B and Solution Let’s Divvy These Up

Problem

1. At Yore Elementary School, each class has between $$15$$ and $$35$$ students. If there are $$78$$ students in Grade $$6$$, and all Grade $$6$$ classes have the same number of students, how many Grade $$6$$ classes are there?

2. At Hyz Elementary School, there is the same number of Grade $$6$$ classes as at Yore, but there are $$84$$ Grade $$6$$ students. If all Grade $$6$$ classes have the same number of students, how many students are in each Grade $$6$$ class?

3. For each of the numbers in the first column of the table given below, first determine if it could be the total number of students in three very large classes of equal size, and then calculate the digit sum of the number. Do you notice any connection between these two things?
Note: The digit sum of a number is the sum of its digits. For example, the digit sum of $$63$$ is $$6+3=9$$.

Number Could be total? Digit Sum
$$1008$$
$$1023$$
$$1741$$
$$2238$$
$$1759$$
$$1902$$

Solution

1. Since the classes are of equal size, the number of classes must divide evenly into $$78$$. The numbers which do so are $$1, 2, 3, 6, 13, 26, 39$$, and $$78$$. The only number that is between $$15$$ and $$35$$ is $$26$$. Since $$3\times 26=78$$, that means there are $$3$$ classes of $$26$$ students.

2. If there are $$84$$ students in $$3$$ classes, then there are $$84\div 3=28$$ students in each class.

Alternatively, since $$84-78=6$$, that means there are $$6\div 3=2$$ more students in each class at Hyz Elementary than at Yore. So there are $$26+2=28$$ students in each class.

3. We can try dividing each number by $$3$$. \begin{aligned} 1008 \div 3 & = 336\\ 1023 \div 3 & = 341\\ 1741 \div 3 & = 580.\bar{3}\\ 2238 \div 3 & = 746\\ 1759 \div 3 & = 586.\bar{3}\\ 1902 \div 3 & = 634\end{aligned} Since $$3$$ divides evenly into $$1008, 1023, 2238,$$ and $$1902$$, these four numbers could be the total number of students in $$3$$ classes of equal size. Since $$3$$ does not divide evenly into $$1741$$ or $$1759$$, these two numbers could not be the total number of students in $$3$$ classes of equal size.

We add this information, along with the digit sum of each number, to the table.

Number Could be total? Digit Sum
$$1008$$ Yes $$9$$
$$1023$$ Yes $$6$$
$$1741$$ No $$13$$
$$2238$$ Yes $$15$$
$$1759$$ No $$22$$
$$1902$$ Yes $$12$$

The digit sums for the numbers $$1008, 1023, 2238,$$ and $$1902$$ are all multiples of $$3$$. These are the numbers that can be divided into $$3$$ equal groups. The digit sums for the other two numbers are not multiples of $$3$$. In fact, a number is a multiple of $$3$$ exactly when its digit sum is also a multiple of $$3$$.