Idea: If a number d divides a number exactly, that number can be written as a multiple of d.
- Given: \(d\mid a\), \(d\mid b\), \(d\mid c\).
- This means there exist integers \(k_1\), \(k_2\), \(k_3\) such that:
\(a = d\,k_1\), \(b = d\,k_2\), \(c = d\,k_3\). - Add the three:
\(a + b + c = d\,k_1 + d\,k_2 + d\,k_3\). - Factor out \(d\):
\(a + b + c = d\,(k_1 + k_2 + k_3)\). - \(k_1 + k_2 + k_3\) is an integer, so \(a + b + c\) is a multiple of \(d\).
- Therefore, \(d\mid (a + b + c)\). So the statement is true.
Quick check (example): Let \(d=5\), \(a=10\), \(b=15\), \(c=25\). Then \(a+b+c=50\), and \(5\mid 50\).