Base conversion

1 Explain why the number 1836.108 for the ratio Rest mass of electron would be suitable for communication with other civilisations whereas neither the rest mass of the proton nor that of the electron would be.

1 In decimal (base 10) form, the number \(N\) is 15260.

1. Express \(N\) in binary (base 2) form.
2. Using the binary form of \(N\), show that \(N\) is divisible by 7 .

5

1. Express as a decimal (base-10) number the base-23 number \(7119 _ { 23 }\).
2. Solve the linear congruence \(7 n + 11 \equiv 9 ( \bmod 23 )\).
3. Let \(N = 10 a + b\) and \(M = a + 7 b\), where \(a\) and \(b\) are integers and \(0 \leqslant b \leqslant 9\).
   1. By considering \(3 N - 7 M\), prove that \(23 \mid N\) if and only if \(23 \mid M\).
   2. Use a procedure based on this result to show that \(N = 711965\) is a multiple of 23 .

1 In this question you must show detailed reasoning. Express the number \(\mathbf { 4 1 7 2 3 } _ { 10 }\) in hexadecimal (base 16).

1 In this question you must show detailed reasoning. The number \(N\) is written as 28 A 3 B in base-12 form. Express \(N\) in decimal (base-10) form.