8.02a Number bases: conversion and arithmetic in base n

2 A civilisation which works in base 5 sends out the first 6 digits of \(\pi\) as 3.032 32. Convert this to base 10.

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 .

8 Let \(f ( n )\) denote the base- \(n\) number \(2121 _ { n }\) where \(n \geqslant 3\).

1. 1. For each \(n \geqslant 3\), show that \(\mathrm { f } ( n )\) can be written as the product of two positive integers greater than \(1 , \mathrm { a } ( n )\) and \(\mathrm { b } ( n )\), each of which is a function of \(n\).
   2. Deduce that \(\mathrm { f } ( n )\) is always composite.
2. Let \(h\) be the highest common factor of \(\mathrm { a } ( n )\) and \(\mathrm { b } ( n )\).
   1. Prove that \(h\) is either 1 or 5 .
   2. Find a value of \(n\) for which \(h = 5\).

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

5 Given that \(n\) is a positive integer greater than 2 , prove that

1. \(\quad 10201 _ { n }\) is a square number.
2. \(\quad 1221 _ { n }\) is a composite number.

4

1. (a) Find all the quadratic residues modulo 11.
   (b) Prove that the equation \(y ^ { 5 } = x ^ { 2 } + 2017\) has no solution in integers \(x\) and \(y\).
2. In this question you must show detailed reasoning. The numbers \(M\) and \(N\) are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that \(M\) is divisible by \(N\).

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.

13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1 \mathrm {~s}\). Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7s.

In this question, the function INT(\(X\)) is the largest integer less than or equal to \(X\). For example, $$\text{INT}(3.6) = 3,$$ $$\text{INT}(3) = 3,$$ $$\text{INT}(-3.6) = -4.$$ Consider the following algorithm. \begin{align} \text{Step 1} \quad & \text{Input } B
\text{Step 2} \quad & \text{Input } N
\text{Step 3} \quad & \text{Calculate } F = N \div B
\text{Step 4} \quad & \text{Let } G = \text{INT}(F)
\text{Step 5} \quad & \text{Calculate } H = B \times G
\text{Step 6} \quad & \text{Calculate } C = N - H
\text{Step 7} \quad & \text{Output } C
\text{Step 8} \quad & \text{Replace } N \text{ by the value of } G
\text{Step 9} \quad & \text{If } N = 0 \text{ then stop, otherwise go back to Step 3} \end{align}

1. Apply the algorithm with the inputs \(B = 2\) and \(N = 5\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. [5]
2. Explain what happens when the algorithm is applied with the inputs \(B = 2\) and \(N = -5\). [4]
3. Apply the algorithm with the inputs \(B = 10\) and \(N = 37\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. What are the output values when \(B = 10\) and \(N\) is any positive integer? [4]

The function INT(\(C\)) gives the largest integer that is less than or equal to \(C\). For example: INT(4.8) = 4, INT(7) = 7, INT(0.8) = 0, INT(−0.8) = −1, INT(−2.4) = −3. Consider the following algorithm. Line 10 \quad Input \(A\) and \(B\) Line 20 \quad Calculate \(C = B \div A\) Line 30 \quad Let \(D =\) INT(\(C\)) Line 40 \quad Calculate \(E = A \times D\) Line 50 \quad Calculate \(F = B - E\) Line 60 \quad Output the value of \(F\) Line 70 \quad Replace \(B\) by the value of \(D\) Line 80 \quad If \(B = 0\) then stop, otherwise go back to line 20

1. Apply the algorithm using the inputs \(A = 10\) and \(B = 128\). Record the values of \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) every time they change. Record the output each time line 60 is reached. [4]
2. Show what happens when the input values are \(A = 10\) and \(B = -13\). [5]