# 2-Digit Square

## Problem

*2-digit Square printable sheet*

A $2$-digit number is squared. When the $2$-digit number is reversed and squared, the difference between the squares is also a square.

## Getting Started

## Student Solutions

The 2-digit number is either $65$ or $56$.

### Proof:

Any 2-digit number can be represented as $10a + b$. We need $(10a+b)^2 - (10b+a)^2 = 99a^2 - 99b^2 =9 \times 11 \times (a^2 - b^2)$ to be a square.

This means that $(a^2 - b^2)$ must be 11 and so $(a - b)(a + b) = 11$ making, $a - b = 1$ and $a + b = 11$. This gives $a = 6$, and $b = 5$.

If we find a solution with $a > b$ then, by reversing the digits, we get a second solution.

## Teachers' Resources

**Why do this problem?**

This problem provides reinforcement of the concept of place value and experience of reading the words in a question and forming an algebraic expression using the information given. It also provides practice in algebra involving the difference of two squares, factorising and solving linear simultaneous equations.

### Possible approach

If you think the class will not remember having learnt the difference of two squares the class could first work on and discuss Plus Minus. However this problem leads naturally into the difference of two squares without the learner having to recognise it at first so it could provide a useful reminder in itself. The learners could first work individually to give them 'thinking time', then work in pairs to support each other and to give an opportunity for mathematical talk, and finally there could be a class discussion.

**Key questions**

Give an example of a 2-digit number . [e.g. 27]

What place value does each digit hold/stand for? [2 tens, 7 units]

Fill in the blank: 27 = 2 times____+ 7

If the digits are reversed what will the new number be? [72]

If a 2 digit number has tens digit a and units digit b then the number is ___times a + ___?

If you know a number is a square what can you say about its factors?

**Possible support**

The problem Plus Minus is a little easier.

**Possible extension**

What's Possible? is another non-standard problem involving the difference of two squares.