Modular arithmetic

Use the Euclidean algorithm to show that 124 and 17 are relatively prime (coprime).

Hence solve the equation

124 x + 17 y = 10

Solve the congruence equation

124 x ≡ 6 mod 17

The polynomial F(x) is a quartic such that

where p, q, r and s are distinct constants.

Determine the number of possible quartics given that

the constants p, q, r and s belong to the set {– 4, –2, 1, 3, 5}

the constants p, q, r and s belong to the set {– 4, –2, 0, 1, 3, 5}

A 3-digit positive integer N = abc has the following properties

• N is divisible by 11

• the sum of the digits of N is even

• N ≡ 8 mod 9

Use the first two properties to show that

a – b + c = 0

Hence determine all possible integers N, showing all your working and reasoning.

In this question you must show detailed reasoning.

Without performing any division, explain why n = 20 210 520 is divisible by 66

Use the Euclidean Algorithm to find integers a and b such that

125 a + 87 b = 1

Hence write down a multiplicative inverse of 87 modulo 125

Solve the linear congruence

87 x ≡ 16 (mod 125)

The four digit number n = abcd satisfies the following properties:

(1) n ≡ 3 (mod 7) (2) n is divisible by 9 (3) the first two digits have the same sum as the last two digits (4) the digit b is smaller than any other digit (5) the digit c is even

Use property (1) to explain why 6a + 2b + 3c + d ≡ 3 (mod 7)

Use properties (2), (3) and (4) to show that a + b = 9

Deduce that c ≡ 5(a – 1) (mod 7)

Hence determine the number n, verifying that it is unique. You must make your reasoning clear.