Use mathematical induction to prove the following statement. If a, c, and n are any integers with n> 1 and a =c (mod n), then for every integer m 2 1, am = cm (mod n). You may use the following theorem in the proof. Theorem 8.4.3(3): For any integers r, s, t, u, and n with n> 1, ifr=s (mod n) and tu (mod n), then rt E su (mod n). Proof by mathematical induction: Let a, c, and n be any integers with n> 1, and suppose that a =c (mod n). Let the property P(m) be the congruence am [We must show that P(m) is true for every integer m 2 1.] Show that P(1) is true: Select P(1) from the choices below. O a' = c (mod m) O a' = c (mod n) O a" = c" (mod m) O 2° c° (mod m) O ° = c° (mod n) The truth of the chosen statement follows by applying one of the laws of exponents to the hypothesis. Show that for each integer k 2 1,-Select- Let k be any integer with k21 and suppose that a= This is the -Select-- We must show that -Select- O ak a * (mod n) O gk + 1 ck+1 (mod n) O a" = c" (mod k) O an +1= c + 1 (mod k) is true. In other words, to complete the inductive step, which of the following must we show to be true? Now a =c (mod n) by assumption, and ak = (mod n) by -Select-- By Theorem 8.4.3(3), we can multiply the left- and right-hand sides of these two congruences together to obtain * (mod n). a. Simplify both sides of the resulting congruence to obtain ak +1= (mod n). Thus, P(k + 1) is true.
Use mathematical induction to prove the following statement. If a, c, and n are any integers with n> 1 and a =c (mod n), then for every integer m 2 1, am = cm (mod n). You may use the following theorem in the proof. Theorem 8.4.3(3): For any integers r, s, t, u, and n with n> 1, ifr=s (mod n) and tu (mod n), then rt E su (mod n). Proof by mathematical induction: Let a, c, and n be any integers with n> 1, and suppose that a =c (mod n). Let the property P(m) be the congruence am [We must show that P(m) is true for every integer m 2 1.] Show that P(1) is true: Select P(1) from the choices below. O a' = c (mod m) O a' = c (mod n) O a" = c" (mod m) O 2° c° (mod m) O ° = c° (mod n) The truth of the chosen statement follows by applying one of the laws of exponents to the hypothesis. Show that for each integer k 2 1,-Select- Let k be any integer with k21 and suppose that a= This is the -Select-- We must show that -Select- O ak a * (mod n) O gk + 1 ck+1 (mod n) O a" = c" (mod k) O an +1= c + 1 (mod k) is true. In other words, to complete the inductive step, which of the following must we show to be true? Now a =c (mod n) by assumption, and ak = (mod n) by -Select-- By Theorem 8.4.3(3), we can multiply the left- and right-hand sides of these two congruences together to obtain * (mod n). a. Simplify both sides of the resulting congruence to obtain ak +1= (mod n). Thus, P(k + 1) is true.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Use mathematical induction to prove the following statement.
If a, c, and n are any integers with n> 1 and a =c (mod n), then for every integer m 2 1, am = cm (mod n).
You may use the following theorem in the proof.
Theorem 8.4.3(3): For any integers r, s, t, u, and n with n> 1, ifr=s (mod n) and tu (mod n), then rt E su (mod n).
Proof by mathematical induction:
Let a, c, and n be any integers with n> 1, and suppose that a E C (mod n). Let the property P(m) be the congruence am
[We must show that P(m) is true for every integer m 2 1.]
Show that P(1) is true: Select P(1) from the choices below.
O a' = c (mod m)
O a' = c (mod n)
O a" = c" (mod m)
O 2° c° (mod m)
O ° = c° (mod n)
The truth of the chosen statement follows by applying one of the laws of exponents to the hypothesis.
Show that for each integer kz1,-Select-
Let k be any integer with k21 and suppose that a
This is the -Select-
We must show that -Select-v is true. In other words, to complete the inductive step, which of the following must we show to be true?
O ak a * (mod n)
O gk + 1 ck+1 (mod n)
O a" =c (mod k)
O an +1= c + 1 (mod k)
Now a = c (mod n) by assumption, and ak = c (mod n) by -Select--
By Theorem 8.4.3(3), we can multiply the left- and right-hand sides of these two congruences together to obtain
* (mod n).
Simplify both sides of the resulting congruence to obtain
ak +1=
(mod n)
Thus, P(k + 1) is true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5515c3ba-7f5d-4ee4-b5cd-03b0cd5507cb%2Ffccb0d8e-5a4c-456e-8f18-e54f9e7e8670%2F606u446_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use mathematical induction to prove the following statement.
If a, c, and n are any integers with n> 1 and a =c (mod n), then for every integer m 2 1, am = cm (mod n).
You may use the following theorem in the proof.
Theorem 8.4.3(3): For any integers r, s, t, u, and n with n> 1, ifr=s (mod n) and tu (mod n), then rt E su (mod n).
Proof by mathematical induction:
Let a, c, and n be any integers with n> 1, and suppose that a E C (mod n). Let the property P(m) be the congruence am
[We must show that P(m) is true for every integer m 2 1.]
Show that P(1) is true: Select P(1) from the choices below.
O a' = c (mod m)
O a' = c (mod n)
O a" = c" (mod m)
O 2° c° (mod m)
O ° = c° (mod n)
The truth of the chosen statement follows by applying one of the laws of exponents to the hypothesis.
Show that for each integer kz1,-Select-
Let k be any integer with k21 and suppose that a
This is the -Select-
We must show that -Select-v is true. In other words, to complete the inductive step, which of the following must we show to be true?
O ak a * (mod n)
O gk + 1 ck+1 (mod n)
O a" =c (mod k)
O an +1= c + 1 (mod k)
Now a = c (mod n) by assumption, and ak = c (mod n) by -Select--
By Theorem 8.4.3(3), we can multiply the left- and right-hand sides of these two congruences together to obtain
* (mod n).
Simplify both sides of the resulting congruence to obtain
ak +1=
(mod n)
Thus, P(k + 1) is true.
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