Reference no: EM13822952
1. Answer each of the following questions, and in each case fully justify your answers.
(a) If x and y are integers, is 9 a factor of 3x . 15y?
(b) Suppose x is an integer such that
2. 3 . 4 . 5 . x = 59 . 58 . 57 . 56 . 55:
(i) Does 59 j x?
(ii) Does 29 j x?
(iii) Does 118 j x?
2. Let a, b, c be any integers. For each of the following statements, if it is true prove it or if it is false provide a counterexample.
(i) If d | a and d | b, then gcd(a, b) = d.
(ii) If a | b and b | c, then c | a.
(iii) If b ≡ 0(mod a) and c ≡ 0(mod b), then c ≡ 0(mod a).
3. Prove that for all positive integers a, b, c and d,
if gcd(ab, c) = d and c j ab, then c = d.
4. Prove the following statement.
For any integer n ≥ 2, n2 - 3 is never divisible by 4.
5. Use the Euclidean algorithm to calculate gcd(672, 150).
Determine whether or not there exists a solution to the following linear Diophantine equation:
672m + 150n = 6:
If a solution exists, give integers m and n that satisfy the equation.
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