Large numbers

I wanted to post this as a problem but decided to put it as a note as I put an extended version of this.

Two numbers, 823519 and 274658, such that if the first number is divided by $x$ , the remainder would be three times the remainder obtained by dividing the second number by $x$ . Find and prove that $x$ has only one value.

#NumberTheory #Sharky

Easy Math Editor

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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

$274658\equiv r\pmod x\stackrel{\times 3}\implies 823974\equiv 3r\pmod x$

$\implies 823974-823519\equiv3r-3r\equiv 455\equiv 0\pmod x\implies x\mid 5\cdot 7\cdot 13$

There's now a limited amount of $x$ 's that could work - only the factors of $5\cdot 7\cdot 13$ , which after bashing (all the $7$ possibilities) gives $\boxed{91}$ as the only solution.

mathh mathh - 6 years, 10 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}`	$\boxed{123}$