Cryptograms Problem

https://brilliant.org/practice/cryptograms-2/?p=7

How did we know that X must be equal to 1?

#Algebra

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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}`	$\boxed{123}$

Comments

If $X$ is any number greater than $2$ , suppose let's say $X = 2$ then we will get it as : $Y + 10Z = 89 \times 2 = 178$ But it is given that $X< Y, Z$ are no-zero digit that means they lie between $(0,10)$ . So, the maximum value of $Y = 9$ or $Y = 8$ (as it is given that all are distinct numbers) and the maximum value of $10Z = 10 \times 9 = 90$ . Now : $max(Y + 10Z) = 8 + 90 = 98 \qquad (or) \qquad 9 + 90 = 99$ If $X = 2$ we will get $Y + 10Z = 178$ but the maximum value of $Y + 10Z = 98 \space (or) \space 99$ and hence there will be a contradiction. So, $X$ must be equal to 1.

Ram Mohith - 2 years, 8 months ago

It helped, Thank You! :)

Devansh Arora - 2 years, 8 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}`	$\boxed{123}$