Question

Show that gcd(a + b, lcm(a, b)) = gcd(a, b) for all a, b ∈ Z.

Answer #1

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(a) If a and b are positive integers, then show that lcm(a, b) ≤
ab.
(b) If a and b are positive integers, then show that lcm(a, b)
is a multiple of gcd(a, b).

1. Show that gcd(1137, -419) = gcd (1137, 419) = gcd (419, 299)
= gcd (299, 120)= gcd (120, 59).
Can you use this to compute gcd(1137, -419)?
2. Show that the gcd(n,n+1)=1 for all n∈Z.
3. Calculate gcd(181451, 186623).

Question 3.
calculate each of the following quantities: (Explain!)
gcd(648, 1083)
lcm(1083, 15435)
lcm(175, 25480)

(a) If a and b are positive integers, then show that gcd(a, b) ≤
a and gcd(a, b) ≤ b.
(b) If a and b are positive integers, then show that a and b are
multiples of gcd(a, b).

Use the Euclidean algorithm to find GCD(221, 85). Draw the Hasse
diagram displaying all divisibilities among the numbers 1, 85, 221,
GCD(85, 221), LCM(85, 221), and 85 × 221.
- Now I already found the gcd and the lcm but I forgot how to
draw the hasse diagram
GCD = 17 and LCM =1105

(a) Show that if gcd(a, m) > 1 that there exists [b] 6= [0]
with [a][b] = [0] (we say that [a] is a zero divisor ). (b) Use
this to show that if gcd(a, m) > 1 then [a]m is not a unit.

Suppose a,b,c belong to Z and gcd(b,c) = 1 . Prove that if
b/(ac), then b/a.

Show that there are infinitely many pairs integers a and b with
gcd(a, b) = 5 and a + b = 65

Prove that for all non-zero integers a and b, gcd(a, b) = 1 if
and only if gcd(a, b^2 ) = 1

Prove that if gcd(a,b)=1 and c|(a+b), then
gcd(a,c)=gcd(b,c)=1.

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