What are the quotient and remainder when (a) -127 is divided by 19?
(b) -1 is divided by 23?
What time does a 24-hour clock read (a) 45 hours before it reads 12:00? (b) 168 hours after it reads 19:00?
Letmbeapositiveinteger. Showthatamodm=bmodmifa≡b(modm).
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List five integers that are congruent to 4 modulo 12.
Show that if a, b, c, and m are integers such that m ≥ 2, c > 0, and a ≡ b (mod m), then ac ≡ bc (mod mc).
1/2 CSI2101/UOttawa/MdH/W22
Prove that for every positive integer n, there are n consecutive composite integers. [Hint: Consider the n consecutive integers starting with (n + 1)! + 2.]
Determine whether 27 − 1 is prime. Problem 8
If the product of two integers is 27 · 38 · 52 · 711 and their greatest common divisor is 23 · 34 · 5, what is their least common multiple?
Show that 937 is an inverse of 13 modulo 2436.
Problem 10
Solve the congruence 2x ≡ 7 (mod 17) using the inverse of 2 modulo 17.
Problem 11
Solve the congruence 34x ≡ 77 (mod 89) using the moduler inverse method.
2/2 CSI2101/UOttawa/MdH/W22
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