Chapter 5, Problem 15CRP

Explanation of Solution

The digit each letter represents:

Consider the addition equation where the alphabets are assigned the traditional 10 base notation, that is, the decimal digits from $0$ to $9$.

$\left(\text{XYZ}\right)+\left(\text{YWY}\right)=\left(\text{ZYZW}\right)$ (1)

Consider the unit place addition with carry to the tens place CR1,

$\text{Z+Y=W+}\left(\text{CR1}\times \text{10}\right)$ (2)

Consider the tens place addition with carry to the hundreds place CR2,

$\text{Y+W+CR1=Z+}\left(\text{CR2}\times \text{10}\right)$ (3)

Consider the hundreds place addition with carry to the thousands place CR3,

$\begin{array}{l}\text{X+Y+}\left(\text{CR2}\times \text{10}\right)\text{=Y+}\left(\text{CR3}\times \text{10}\right)\\ \text{X+}\left(\text{CR2}\times \text{10}\right)\text{=}\left(\text{CR3}\times \text{10}\right)\end{array}$ (4)

Consider the thousands place addition,

CR3=Z (5)

The maximum possible carry in a decimal addition of two numbers is 1. Thus,

$Z=1$

A number repeats itself after every $10$ digits .Thus, in equation (4), Y is repeating, it means,

$\text{X+CR2=10}$

The maximum possible carry in a decimal addition of two numbers is 1 and the maximum possible value for X is $9$