Excel in math, science, and engineering

New user? Sign up

Existing user? Sign in

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 7 months ago

Sort by:

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 \(a \gt 2\). – Brian Charlesworth · 7 months ago

Log in to reply

@Brian Charlesworth – Furthermore, we have \(b\equiv 10\pmod{13}\) – Prasun Biswas · 5 months, 3 weeks ago

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

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 \(a \gt 2\). – Brian Charlesworth · 7 months ago

Log in to reply

– Prasun Biswas · 5 months, 3 weeks ago

Furthermore, we have \(b\equiv 10\pmod{13}\)Log in to reply