System of Congruence problem

Find all solutions to the system of congruences: x ≡ 2 (mod 3) x ≡ 1 (mod 4) x ≡ 3 (mod 5)

Fermat's little theorem

Use Fermat's Little Theorem to compute 3^302 mod 5, 3^302 mod 7, and 3^302 mod 11. Use your results to find 3 ^ 302 mod 385 (note 385 = 5 . 7 . 11.)

Show that the binary expansion of a positive integer can be obtained from its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits.

Show that the binary expansion of a positive integer can be obtained from its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits.

Give a procedure for converting from the hexadecimal expansion of an integer to its octal expansion using binary notation as an intermediate step.

Give a procedure for converting from the hexadecimal expansion of an integer to its octal expansion using binary notation as an intermediate step.

Ordered Pais & Sets

1. a) Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an example that we cannot define the ordered triple (x, y, z) as the set {{x}, {x,y}, {x,y,z}} 2. Which of the following are binary or unary operations on the given sets? For those that are not, where do they fail? a) x ◦ y = 1/x if x is positive S = set of all real numbers 1/(-x) if x is negative b) x ◦ y = xy (concatenation); S = se ... click for more