Question

(d)n- 1013 2. Let a, b, c, d be integers. Prove the statement or give a counterexample (a) If (ab) c, then a |c and alc. (b) If a l b and c|d, then ac bod (c) If aYb and alc, then aYbc. (d) If a31b4, then alb. (e) If ged(a, b) 1 and alc and b c, then (ab) c. Here a and b are relatively prime integers, also called coprime integers.] rherF and r is an integer with 0r<

Answer 1

(a) If ab vert c then a vert c and b vert c is not true. counter example: 75/3 75 = 25 times 3 rightwardsdoublearrow 25/5 and 3/3 thus when a = 25 b = 3 c = 3 then if ab vert c rightwardsdoublearrow a vert c and b vert c b) a vert b rightwardsdoublearrow b = ak_1 where k_1 elementof Z. c vert d rightwardsdoublearrow d = c k_2 where k_2 elementof Z consider bd = (ak_1 (ck_2) = sc k_1 k_2 = ac(k) If k_1, k_2 elementof Z then k_1, k_2 elementof Z say k_1 k_2 = k elementof Z rightwardsdoublearrow ac/bd

In part (c).

The given statement in part (c) is disprove.

Counter example:

a = 6, b =3 and c = 4

6 does not divide 3.

6 does not divide 4.

But 6/4*3 => 6/12 .that is 6 divides 12.

Hence given statement is disprove.

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d) $a^{3}/b^{4}$ then a/b.

Let us consider a does not divides b.

Which gives a^3 does not divides b^3

Which gives a^3 does not divides b^3*b (since a^3 dosent divide b^3 and a does not divide b)

Which gives a^3 does not divides b^4.

But it is given that a^3 divides b^4.

Thus a/b.

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e)

Given that gcd (a,b)=1 and a /c and b /c then ab/c.

By using definition of divisibility,

a/c => c = ax' ...........(1)

b/c => c = bx'' ...........(2) [x' and x'' are integer]

gcd(a,b) = 1 gives

ax+by = 1 x, y are integers.

Multiply both sides by c.

acx+bcy = c

a(bx'')+b(ax') = c

abx''+abx' = c

ab(x'+x'') = c (x'+x'') is integer.

By divisibility, ab/c.