# Problem of the Week Problem C and Solution Where Does the Year Go?

## Problem

The positive integers are written consecutively in rows, with seven integers in each row. That is, the first row contains the integers $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, and $$7$$. The second row contains the integers $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, and $$14$$. The third row contains the integers $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, $$20$$, and $$21$$, and so on.

Determine the row and the column that the integer $$2024$$ is in.

## Solution

The last number in row $$1$$ is $$7$$, the last number in row $$2$$ is $$14$$, and the last number in row $$3$$ is $$21$$. Observe that the last number in each row is a multiple of $$7$$. Furthermore, the last number in row $$n$$ is $$7\times n$$. So, we will find the largest multiple of $$7$$ that is less than $$2024$$.

We solve the equation $$7\times n = 2024$$ to get $$n \approx 289.14$$.

Therefore, the largest multiple of $$7$$ that is less than 2024 is $$289 \times 7 = 2023$$. This means that $$2023$$ is the last number in row $$289$$. Thus, $$2024$$ will be the first number in in row $$290$$.

Therefore, $$2024$$ is in row $$290$$ and column $$1$$.