Problem
NT-B2-M11-P003 The Last Three Digits
#3
★★☆☆☆ Level 2 of 5
Prove that a decimal number is divisible by \(8\) if and only if the number formed by its last three digits is divisible by \(8\). Find all digits \(c\) for which \(\overline{45c2}\) is divisible by \(8\).
Separate the last three digits: \(N=1000q+r\).
Let \(N=1000q+r\), where \(0\le r<1000\) is the number formed by the last three digits. Since \(1000\) is divisible by \(8\), we have \(N\equiv r\pmod8\).
For \(\overline{45c2}\), the last three digits form \(502+10c\). We need \(502+10c\equiv0\pmod8\), or \(6+2c\equiv0\pmod8\). Thus \(2c\equiv2\pmod8\), so \(c\equiv1\pmod4\). The answer is \(c=1,5,9\).
It is useful to emphasise the difference between modulus \(8\) and modulus \(10^3\).