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5. (Euclidean algorithm practice: integer solutions) The following problems again use the notation $(m, n)$ to denote greatest common divisor of m and n. a) Use the Euclidean algorithm to compute (23, 41). b) Use reverse substitutions from equations produced by the above Euclidean al- gorithm to find a solution $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ to the equation $23x + 41y = 1$. c) Describe the entire set of solutions $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ such that $23x + 41y = 1$. d) Find a solution $x \in \{0,..., 40\}$ to the congruence $23x \equiv 1 \pmod{41}$. e) Find a solution $x \in \{0,..., 40\}$ to the congruence $23x \equiv 7 \pmod{41}$.

          5. (Euclidean algorithm practice: integer solutions) The following problems again use
the notation $(m, n)$ to denote greatest common divisor of m and n.
a) Use the Euclidean algorithm to compute (23, 41).
b) Use reverse substitutions from equations produced by the above Euclidean al-
gorithm to find a solution $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ to the equation $23x + 41y = 1$.
c) Describe the entire set of solutions $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ such that $23x + 41y = 1$.
d) Find a solution $x \in \{0,..., 40\}$ to the congruence $23x \equiv 1 \pmod{41}$.
e) Find a solution $x \in \{0,..., 40\}$ to the congruence $23x \equiv 7 \pmod{41}$.

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5. (Euclidean algorithm practice: integer solutions) The following problems again use
the notation (m, n) to denote greatest common divisor of m and n.
a) Use the Euclidean algorithm to compute (23, 41).
b) Use reverse substitutions from equations produced by the above Euclidean al-
gorithm to find a solution (x, y) ∈ℤ×ℤ to the equation 23x + 41y = 1.
c) Describe the entire set of solutions (x, y) ∈ℤ×ℤ such that 23x + 41y = 1.
d) Find a solution x ∈{0,..., 40} to the congruence 23x ≡ 1 41.
e) Find a solution x ∈{0,..., 40} to the congruence 23x ≡ 7 41.

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Calculus: Early Transcendentals

James Stewart 8th Edition

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(Euclidean algorithm practice: integer solutions) The following problems again use the notation (m,n) to denote greatest common divisor of m and n. a) Use the Euclidean algorithm to compute (23,41). b) Use reverse substitutions from equations produced by the above Euclidean algorithm to find a solution (x,y)inZ imes Z to the equation 23x+41y=1. c) Describe the entire set of solutions (x,y)inZ imes Z such that 23x+41y=1. d) Find a solution xin{0,dots,40} to the congruence 23x-=1(mod41). e) Find a solution xin{0,dots,40} to the congruence 23x-=7(mod41). 5. (Euclidean algorithm practice: integer solutions) The following problems again use the notation (m,n) to denote greatest common divisor of m and n. a) Use the Euclidean algorithm to compute (23,41. b Use reverse substitutions from equations produced by the above Euclidean al- gorithm to find a solution (x,y e Z Z to the equation 23x+4ly=1. cDescribe the entire set of solutions, Z Zsuch that 23x+41y=1 d Find a solution {0....40} to the congruence 23=1mod41 e Find a solution x {0,...,40} to the congruence 23x =7(mod 41.

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