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# 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.

Note by Sharky Kesa
3 years, 5 months ago

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

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

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.

- 3 years, 5 months ago