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