When the positive integers \(p\), \(q\) and \(r\) are divided by \(13\), the respective remainders are \(10\), \(9\) and \(7\). Show that \(p+q+r\) is divisible by \(13\).
\(p+q+r\) is divisible by \(13\).
Using Euclid's division lemma:
\(p = \)\(q_1 +\) - - - - (I)
\(q =\)\(q_2 +\) - - - - (II)
\(r =\)\(q_3 + \) - - - - (III)
Add (I), (II) and (III).
\(p+q+r=\) \((q_1 + q_2 + q_3 +\)\()\)
Thus, \(p+q+r\) is divisible by \(13\).
Hence, we proved.