[modern algebra] Use sentences to explain what you are doing. 2. (10 points) TRU

Question

[modern algebra] Use sentences to explain what you are doing.

2. (10 points) TRUE or FALSE (If a statement is FALSE, give an example that shows it is false.) (a) Suppose a, b, care integers and a 0. If ab = ac (mod 10), then b = c (mod 10). (b) Every integer except zero is prime or a product of primes. (c) Let a, b, c E Z. If a b (mod 4) and b c (mod 4) then a c (mod 8). (d) Let a, b, c E Z. If a (mod 4) and b c (mod 8) then a (mod 4). (e) Ifa and b are nonzero integers and there are integers u and u such that au + bu = 2, then the GCD of a and b is 2.

Explanation / Answer

a) False

2*3 (mod 8) = 2*7 (mod 8)
but not 3 = 7 (mod 8)

b) False

1 is neither prime nor is it product of prime

c) False

Let a=3, b=3 and c=15

then a=b(mod 4), b=c(mod 4) but a is not equal to c(mod 8)

d) True

e) False

Let a=6, b=5, u=2 and v=-2

then au+bv = 2

but GCD(a,b) i.e. GCD(6,5) is not 2

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