Download Math 373 Homework 4

Math 373 Homework 4
Due on Feb 18
1.Find all the integers x such that 9x ≡ 8 (mod 7).
Solution. Since 9 ≡ 2 (mod 7) and 8 ≡ 1 (mod 7), the congruence
equation can be restated as
2x ≡ 1
(mod 7).
Now observe that the “reciprocal” of 2 modulo 7 is 4, in that 2 ∗ 4 =
8 ≡ 1 (mod 7). So we have
8x ≡ 4
(mod 7)
x≡4
(mod 7).
which is
So the solutions to the congruence equation are those integers congruent to 4 modulo 7.
2.Which of the following are complete residue systems modulo 11?
Justify your answers.
(a) 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
(b) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
(c) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22
(d) -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Solution. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the standard complete residue
system modulo 11. So we only need to check whether the given list is
equivalent to this one.
(a) Yes . 16 ≡ 5 (mod 11), 32 ≡ 5 ∗ 2 ≡ 10 (mod 11), 64 ≡ 10 ∗ 2 ≡ 9
(mod 11), 128 ≡ 9 ∗ 2 ≡ 7 (mod 11), 256 ≡ 7 ∗ 2 ≡ 3 (mod 11),
512 ≡ 3 ∗ 2 ≡ 6 (mod 11).
(b) Yes . 11 ≡ 0 (mod 11), 13 ≡ 2 (mod 11), 15 ≡ 4 (mod 11),
17 ≡ 6 (mod 11), 19 ≡ 8 (mod 11), 21 ≡ 10 (mod 11).
(c) Yes . 12 ≡ 1 (mod 11), 14 ≡ 3 (mod 11), 16 ≡ 5 (mod 11),
18 ≡ 7 (mod 11), 20 ≡ 9 (mod 11), 22 ≡ 0 (mod 11).
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(d) Yes . −5 ≡ 6 (mod 11), −4 ≡ 7 (mod 11), −3 ≡ 8 (mod 11),
−2 ≡ 9 (mod 11), −1 ≡ 10 (mod 11).
3.Which of the following are reduced residue systems modulo 18? Justify your answers.
(a) 1, 5, 25, 125, 625, 3125
(b) 5, 11, 17, 23, 29, 35
(c) 1, 25, 49, 121, 169, 289
(d) 1, 5, 7, 11, 13, 17
Solution. By removing numbers that are not coprime with 18 in the
standard complete residue system modulo 18, we know {1, 5, 7, 11,
13, 17} is a reduced residue system modulo 18. It suffices to check
whether each list is equivalent to this one.
(a) Yes . 25 ≡ 7 (mod 18), 125 ≡ 7 ∗ 5 ≡ 17 ≡ −1 (mod 18), 625 ≡
−1 ∗ 5 ≡ 13 (mod 18), 3125 ≡ 13 ∗ 5 ≡ 11 (mod 18).
(b) No , because 5 ≡ 23 (mod 18).
(c) No , because 25 ≡ 169 (mod 18).
(d) Yes .
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