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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 \(a\).

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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 \(a \gt 2\).

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

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TopNewestIf \(4b - 1\) 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 \(a \gt 2\).

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

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