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# Number Theory

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When N is divided by 10, the remainder is a. When N is divided by 13, the remainder is b. If a = 5 and b = $$10$$, what is N modulo 130?

Apr 16, 2022

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Here we have the conditions $$\begin{cases}N\equiv 5 \pmod {10}\\N\equiv 10\pmod{13}\end{cases}$$.

One way without using Chinese Remainder Theorem is to list all numbers with $$N \leq 130$$ and $$N \equiv 10 \pmod{13}$$.

You can start from 13 + 10 = 23, and add 13 each time you progress in the list.

The list is {23, 36, 49, 62, 75, ...}.

Now, note that 75 also satisfies $$N \equiv 5 \pmod{10}$$. Then this is the answer we want.

Note that $$75 \operatorname{mod} 130 = \boxed{75}$$. That's the answer.

Remark: This method is not rigorous at all. For a solution with more rigor, you would need Chinese Remainder Theorem.

Apr 19, 2022