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Let $$a$$ and $$b$$ be integers with $$a>2$$. If $$4b-1$$ and $$5b+2$$ are both multiples of $$a$$, find the value of $$a$$.

Note by D B
1 year, 1 month ago

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If $$4b - 1$$ and $$5b + 2$$ are both multiples of $$a$$, then any integral linear combination of these terms will also be a multiple of $$a$$. We can then state that

$$4(5b + 2) - 5(4b - 1) = na \Longrightarrow 13 = na \Longrightarrow a = 13$$,

since $$13$$ is prime and $$a \gt 2$$.

- 1 year, 1 month ago

Furthermore, we have $$b\equiv 10\pmod{13}$$

- 11 months, 4 weeks ago