1.
Give the definition for these terms. Be sure to set up any notation that is required.
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base \(b\) expansion
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binary
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decimal
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hexadecimal
Exercises 5.2.6 Exercises
2.
3.
Watch the following videos
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How high can you count on your fingers?by James Tanton posted by TED-Ed ? (Spoiler: much higher than 10)
2
youtu.be/UixU1oRW64Q
and learn to count in binary on your hands.
4.
Compute the integer representing the ASCII encoding of these messages.
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apple
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Radiohead
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discrete math
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secret
5.
These integers represent messages encoded in ASCII as described in Subsection 5.2.2. Decode these messages.
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\(\displaystyle 6582119\)
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\(\displaystyle 280991720293\)
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\(\displaystyle 311107740793\)
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\(\displaystyle 22107779118197813113556726561\)
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The following is an integer that is too long to fit in a line.\begin{gather*} 1971486178880874921204775823582727213754122745\\ 127854714387549425398830 \end{gather*}
6.
Compute \(12^{321} \bmod{456}\text{.}\) Follow the steps below if you get stuck.
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Check that you got \(321 = (101000001)_2\text{.}\)
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Compute the successive powers \(p_i\text{,}\) remembering to reduce mod \(456\text{.}\) Did you notice anything that helps this computation go faster?
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Multiply the correct terms together to compute \(12^{321} \bmod{456}\text{.}\)
7.
Compute \(7^{447} \bmod{645}\) using fast exponentiation. Show your work.
8.
Compute \(3^{447} \bmod{645}\) using fast exponentiation. Show your work.
9.
Convert the decimal expansion of each of these integers to a binary expansion. that is , Convert base \(10\) to base \(2\text{.}\)
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\(\displaystyle 1\)
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\(\displaystyle 156\)
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\(\displaystyle 765\)
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\(\displaystyle 23\)
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\(\displaystyle 245\)
10.
Convert the decimal expansion of each of these integers to a hexadecimal expansion. that is , Convert base \(10\) to base \(16\text{.}\)
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\(\displaystyle 123\)
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\(\displaystyle 45326\)
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\(\displaystyle 12\)
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\(\displaystyle 157\)
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\(\displaystyle 149987\)
11.
Convert the binary expansion of each of these integers to a decimal expansion. that is , Convert base \(2\) to base \(10\text{.}\)
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\(\displaystyle (10011)_2\)
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\(\displaystyle (111)_2\)
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\(\displaystyle (101010)_2\)
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\(\displaystyle (111000)_2\)
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\(\displaystyle (1011011)_2\)
12.
Convert the hexadecimal expansion of each of these integers to a decimal expansion. that is , Convert base \(16\) to base \(10\text{.}\)
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\(\displaystyle (A123B)_{16}\)
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\(\displaystyle (81C)_{16}\)
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\(\displaystyle (ABBA)_{16}\)
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\(\displaystyle (DA3)_{16}\)
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\(\displaystyle (253)_{16}\)
13.
Prove that a positive integer is divisible by \(5\) if and only if the last digit is divisible by \(5\text{.}\)
14.
Prove that a positive integer is divisible by \(3\) if and only if the sum of its decimal digits is divisible by \(3\text{.}\)
15.
Prove that a positive integer is divisible by \(11\) if and only if the alternating sum of its decimal digits is divisible by \(11\text{.}\)
