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Homework 4
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(Transitivity rule for modular arithmetic) Let
$a$ ,$b$ ,$c$ , and$n$ be integers, and suppose that$a\equiv b \pmod{n}$ and$b\equiv c \pmod{n}$ . Show that$a\equiv c \pmod{n}$ .Do this in the style of Example 3.3.3, Example 3.3.6, etc., that is, directly from the definitions.
Note: the problem can also be solved in two lines as follows. The idea is not to do this.
calc a ≡ b [ZMOD n] := h1 _ ≡ c [ZMOD n] := h2
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Let
$t$ be an integer for which$t\equiv 2\pmod{4}$ . Show that$3(t ^ 2 + t - 8) \equiv 3(2 ^ 2 + 2 - 8)\pmod{4}$ .Do this in the style of Example 3.3.11, that is by applying the relevant lemmas from Section 3.3 one by one. Note: the problem can also be solved in one line as
rel [h]
. The idea is not to do this.Write this problem only in Lean, not on paper.
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Let
$a$ be an integer for which$a\equiv 3\pmod{5}$ . Show that$a^3+4a^2+3\equiv 1\pmod{5}$ . -
Show that there exists an integer
$k$ , such that$k>50$ and$k\equiv 2\pmod{5}$ and$5k\equiv 6\pmod{8}$ . -
Let
$x$ be an integer. Show that$x^5\equiv x\pmod{5}$ . -
Let
$n$ be an integer which is divisible by 5 and by 12. Show that it is also divisible by 60.