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The point of this problem is to show that some of the important properties of modular arithmetic and coding theory actually come from group theory. Recall that U(n) = {1 ? k < n | gcd(k, n) = 1} is the group of units under multiplication mod n. Let ?(n) = |U(n)|, so that ?(n) is the number of positive integers less than n that are relatively prime to n. a. Fill out the table n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 ?(n) | | | | | | | | | | | b. Prove that a^?(n) ? 1(mod n). (Hint: use a corollary of Lagrange's Theorem). c. Demonstrate that b works for n = 12 and each a ? U(12). d. Use part (b) to prove the following, which is called Fermat's Little Theorem: a^p ? a(mod p) for a prime number p. e. Demonstrate Fermat's Little Theorem for p = 7 and each number in U(7).

          The point of this problem is to show that some of the important properties of modular arithmetic and coding theory actually come from group theory.

Recall that U(n) = {1 ? k < n | gcd(k, n) = 1} is the group of units under multiplication mod n. Let ?(n) = |U(n)|, so that ?(n) is the number of positive integers less than n that are relatively prime to n.

a. Fill out the table

n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
?(n) | | | | | | | | | | |

b. Prove that a^?(n) ? 1(mod n). (Hint: use a corollary of Lagrange's Theorem).

c. Demonstrate that b works for n = 12 and each a ? U(12).

d. Use part (b) to prove the following, which is called Fermat's Little Theorem: a^p ? a(mod p) for a prime number p.

e. Demonstrate Fermat's Little Theorem for p = 7 and each number in U(7).

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The point of this problem is to show that some of the important properties of modular arithmetic and coding theory actually come from group theory.

Recall that U(n) = 1 ? k < n | gcd(k, n) = 1 is the group of units under multiplication mod n. Let ?(n) = |U(n)|, so that ?(n) is the number of positive integers less than n that are relatively prime to n.

a. Fill out the table

n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
?(n) | | | | | | | | | | |

b. Prove that a^?(n) ? 1(mod n). (Hint: use a corollary of Lagrange's Theorem).

c. Demonstrate that b works for n = 12 and each a ? U(12).

d. Use part (b) to prove the following, which is called Fermat's Little Theorem: a^p ? a(mod p) for a prime number p.

e. Demonstrate Fermat's Little Theorem for p = 7 and each number in U(7).

Added by Veronica B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/09/2023

Step 1

U(2) = {1}, o(2) = 1 U(3) = {1, 2}, o(3) = 2 U(4) = {1, 3}, o(4) = 2 U(5) = {1, 2, 3, 4}, o(5) = 4 U(6) = {1, 5}, o(6) = 2 U(7) = {1, 2, 3, 4, 5, 6}, o(7) = 6 U(8) = {1, 3, 5, 7}, o(8) = 4 U(9) = {1, 2, 4, 5, 7, 8}, o(9) = 6 U(10) = {1, 3, 7, 9}, o(10) = 4 U(11) Show more…

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The point of this problem is to show that some of the important properties of modular arithmetic and coding theory actually come from group theory. Recall that U(n) = {1 ≤ k < n | gcd(k, n) = 1} is the group of units under multiplication mod n. Let ϕ(n) = |U(n)|, so that ϕ(n) is the number of positive integers less than n that are relatively prime to n. a. Fill out the table n: 2 3 4 5 6 7 8 9 10 11 12 b. Prove that a^ϕ(n) ≡ 1(mod n). (Hint: use a corollary of Lagrange's Theorem). c. Demonstrate that b works for n = 12 and each a ∈ U(12). d. Use part (b) to prove the following, which is called Fermat's Little Theorem: a^p ≡ a(mod p) for a prime number p. e. Demonstrate Fermat's Little Theorem for p = 7 and each number in U(7).

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