TRUE or FALSE. If the statement is false, give an example showing that it is fal

Question

TRUE or FALSE. If the statement is false, give an example showing that it is false. Let a.b.c Z with a, b 0. If a|b and b|c then a|c. The only divisors of 6 are 1.2,3, and 6. The smallest positive integer in the set {6u + 9v : u, v Z} is 1. Let a an integer. Then the greatest common divisor of a and 0 is |a|. Let a, b, c belongs to Z with a 0 and (b, c) = 1. If a divides the product bc, then a must divide b or a must divide c. Find the quotient and remainder when - 463 is divided by 12. Use the Euclidean Algorithm to answer the following questions Find the greatest common divisor, d, of 25326 and 4211. Find integers u and v such that d = 25326u + 4211 v. Let a be an integer. Prove that the square of a can always be written in the form 5k, 5k + 1, or 5k + 4 for some integer k. Let a, b, c be nonzero integers. Suppose a(b + c) and (b, c) = 1. Prove that (a, b) = 1. Hint: Use the fact that the gcd of a and b is the smallest positive integer that can be written in the form ax + by for integers x and y.

Explanation / Answer

a)true

Claim. If a/b and b/c, then a/c

Proof.

Since a/b, there exists q 2 Z such that b = q a. Similarly, since b/c, there exists r 2 Z

such that c = r b Thus c = r (q a) = (r q) a (by associativity) so a/c.

b)true

Prove of disprove: If a/(b + c), then a/b or a/c.

Proof. The above statement is false and we disprove it by a counterexample (any other method

of proof is probably not the best idea, and more work than necessary if at all possible): For

example, 2/(1 + 5) but 2 does nt divide1 and 2 does not divide 5.

c)false

We can use the Euclidean algorithm to determing (6; 9) = 3. Thus by Hungerford,

theorem we have that 3 is the smallest positive integer in this set.

d)true

e)true

THEOREM (Euclid

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