Help with a cryptogram problem

I need help understanding a problem's solution. Let's say I am given the cryptogram XX + YY + ZZ = XYZ. I know what the variables are (X=1, Y=9, Z=8), but how did those variables equal those amounts? This is the last problem in the first module of the Mathematical Fundementals course, but the explanation they gave me made no sense to me for some reason. Can someone help me explain the solution to this problem in a simpler way?

#Algebra

Easy Math Editor

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Comments

I believed that you're referring to this problem.

Here's an alternative solution:

The largest possible value of the sum of $XX, YY, ZZ$ is $99 + 88 + 77 = 264$ . And since the final sum is a 3-digit number, then $X = 1$ or $X = 2$ only.

Suppose $X = 2$ , the equation simplifies to $22 + YY + ZZ = 2YZ$ . Converting $YY, ZZ, 2YZ$ into proper algebraic expressions gives $10Y + Y, 10Z + Z, 200 + 10Y + Z$ .

Substituting them gives $22 + (10Y + Y) + (10Z + Z) = 200 + 10Y + Z$ . Simplifies it gives $Y + 10Z = 178$ . Since $Y$ and $Z$ are distinct single digits, then there's no solution.

Hence, $X = 1$ only. The equation simplifies to $11 + YY + ZZ = 1YZ$ . Converting $YY, ZZ, 2YZ$ into proper algebraic expressions gives $10Y + Y, 10Z + Z, 100 + 10Y + Z$ .

Substituting them gives $11 + (10Y + Y) + (10Z + Z) = 100 + 10Y + Z$ . Simplifies it gives $Y + 10Z = 89$ . Since $Y$ and $Z$ are distinct single digits, then $Y = 9, Z = 8$ .

And we're done!

Pi Han Goh - 2 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$