1. Use Fermat's theorem to verify that 17 divides 11104 + 1.
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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![1. Use Fermat's theorem to verify that 17 divides 1104 + 1.
2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35).
[Hint: From Fermat's theorem a = 1 (mod 7) and a
(b) If gcd(a, 42) = 1, show that 168 = 3-7.8 divides a- 1.
(c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133|a18 – b18.
3. From Fermat's theorem deduce that, for any integer n > 0, 13|11127+6 +1.
4. Derive each of the following congruences:
(a) a = a (mod 15) for all a.
[Hint: By Fermat's theorem, a' = a (mod 5).]
(b) a' = a (mod 42) for all a.
(c) a13
(d) a' = a (mod 30) for all a.
5. If gcd(a, 30) = 1, show that 60 divides at + 59.
6. (a) Find the units digit of 3 by the use of Fermat's theorem.
(b) For any integer a, verify that a and a have the same units digit.
7. If 7 a, prove that eithera'+1 or a'- 1 is divisible by 7.
[Hint: Apply Fermat's theorem.]
8. The three most recent appearances of Halley's comet were in the years 1835, 1910
1986; the next occurrence will be in 2061. Prove that
= 1 (mod 5).]
= a (mod 3 -7·13) for all a.
4
1010
2061
Zoom out
rton-Elementary..., McGraw-Hill 5th edition).pdf
at x =a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F048faa6b-a100-4249-8bc6-0274c2c51311%2Ff657c86d-4966-4fad-9790-292a3da72beb%2Fursccs9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Use Fermat's theorem to verify that 17 divides 1104 + 1.
2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35).
[Hint: From Fermat's theorem a = 1 (mod 7) and a
(b) If gcd(a, 42) = 1, show that 168 = 3-7.8 divides a- 1.
(c) If gcd(a, 133) = gcd(b, 133) = 1, show that 133|a18 – b18.
3. From Fermat's theorem deduce that, for any integer n > 0, 13|11127+6 +1.
4. Derive each of the following congruences:
(a) a = a (mod 15) for all a.
[Hint: By Fermat's theorem, a' = a (mod 5).]
(b) a' = a (mod 42) for all a.
(c) a13
(d) a' = a (mod 30) for all a.
5. If gcd(a, 30) = 1, show that 60 divides at + 59.
6. (a) Find the units digit of 3 by the use of Fermat's theorem.
(b) For any integer a, verify that a and a have the same units digit.
7. If 7 a, prove that eithera'+1 or a'- 1 is divisible by 7.
[Hint: Apply Fermat's theorem.]
8. The three most recent appearances of Halley's comet were in the years 1835, 1910
1986; the next occurrence will be in 2061. Prove that
= 1 (mod 5).]
= a (mod 3 -7·13) for all a.
4
1010
2061
Zoom out
rton-Elementary..., McGraw-Hill 5th edition).pdf
at x =a
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