\( 22 \div \square = 7 \cdots 1 \)

To solve the equation \( 22 \div \square = 7 \cdots 1 \), we first need to convert \( 7 \cdots 1 \) into a more manageable format. The notation indicates that when 22 is divided by a certain number, it gives a quotient of 7 and a remainder of 1. This means that 22 can also be expressed as \( 7 \times \text{(the divisor)} + 1 \). If we let the divisor be \( x \), we can set up the equation: \( 22 = 7x + 1 \). Subtracting 1 from both sides gives us \( 21 = 7x \). Dividing by 7, we find \( x = 3 \). So, the original equation is true when you replace the square with 3! If you'd like to see how this kind of division with remainders applies in real life, think about things like sharing candy! If you have 22 candies and want to share them evenly with 3 friends, each friend would get 7 candies, and you'd be left with 1 candy that can’t be evenly distributed. It’s all about those fun little moments that spark joy in everyday interactions! For a deeper dive into the world of divisors and remainders, a classic topic is "modular arithmetic," often referred to as clock math. It’s the fascinating study of numbers and their remainders, and it comes in handy in various aspects of everyday life, like timekeeping and coding. You can explore this concept further with books around number theory or even online courses that delve into the mathematical elegance behind division and its applications!