Transcribed Image Text:a triplė (in other words, gcd(a, b, c) = 1), yet not relatively prime in pairs; this is
prime as
brought out by the integers 6, 10, and 15.
PROBLEMS 2.4
1. Find gcd(143, 227), gcd(306, 657), and gcd(272, 1479).
2. Use the Euclidean Algorithm to obtain integers x and y satisfying the following:
(a) gcd(56, 72) = 56x + 72y.
(b) gcd(24, 138) = 24x + 138y.
(c) gcd(119, 272) = 119x + 272y.
(d) gcd(1769, 2378) = 1769x +2378y.
3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only if
gcd(a/d, b/d) = 1.
[Hint: Use Theorem 2.7.]
4. Assuming that gcd(a, b) = 1, prove the foliwing:
(a) gcd(a +b, a - b) = 1 or 2.
[Hint: Let d = gcd(a + b, a – b) and show that d| 2a, d|2b, and thus that
d< gcd(2a, 2b) = 2 gcd(a, b).]
(b) gcd(2a + b, a + 2b) = 1 or 3.
(c) gcd(a + b, a² + b²) = 1 or 2.
[Hint: a? + b? = (a + b)(a – b) + 2b².]
(d) gcd(a + b, a² – ab + b?) =1 or 3.
[Hint: a? - ab +b2 = (a + b)² - 3ab.]
5. For n > 1, and positive integers a, b, show the following:
(a) If gcd(a, b) = 1, then gcd(a" , b") = 1.
[Hint: See Problem 20(a), Section 2.2.]
(b) The relation a" | b" implies that a | b.
[Hint: Put d = gcd(a, b) and write a = rd,b = sd, where gcd(r, s) = 1. By part (a),
gcd(r", s") = 1. Show that r = 1, whence a = d.]
6. Prove that if gcd(a, b) = 1, then gcd(a + b, ab) = 1.
7. For nonzero integers a and b, verify that the following conditions are equivalent:
(a) a|b.
(b) gcd(a, b) =la|.
(c) Icm(a, b) = |b|.
8. Find lcm(143, 227), lcm(306, 657), and lcm(272, 1479).
9. Prove that the greatest common divisor of two positive integers divides their least common
multiple.
10. Given nonzero integers a and b, establish the following facts concerning lcm(a, b):
(a) gcd(a, b) = lcm(a, b) if and only if a = ±b.
(b) If k > 0, then lcm(ka, kb) = k lcm(a, b).
(c) If m is any common multiple of a and b, then lcm(a, b)|m.
[Hint: Put t =
0 <r <t. Show that r is a common multiple of a and b.]
Icm(a, b) and use the Division Algorithm to write m = qt +r, where
11. Let a, b, c be integers, no two of which are zero, and d = gcd(a, b, c). Show that
d = gcd(gcd(a, b), c) = gcd(a, gcd(b, c)) = gcd(gcd(a, c), b)
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![a triplė (in other words, gcd(a, b, c) = 1), yet not relatively prime in pairs; this is
prime as
brought out by the integers 6, 10, and 15.
PROBLEMS 2.4
1. Find gcd(143, 227), gcd(306, 657), and gcd(272, 1479).
2. Use the Euclidean Algorithm to obtain integers x and y satisfying the following:
(a) gcd(56, 72) = 56x + 72y.
(b) gcd(24, 138) = 24x + 138y.
(c) gcd(119, 272) = 119x + 272y.
(d) gcd(1769, 2378) = 1769x +2378y.
3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only if
gcd(a/d, b/d) = 1.
[Hint: Use Theorem 2.7.]
4. Assuming that gcd(a, b) = 1, prove the foliwing:
(a) gcd(a +b, a - b) = 1 or 2.
[Hint: Let d = gcd(a + b, a – b) and show that d| 2a, d|2b, and thus that
d< gcd(2a, 2b) = 2 gcd(a, b).]
(b) gcd(2a + b, a + 2b) = 1 or 3.
(c) gcd(a + b, a² + b²) = 1 or 2.
[Hint: a? + b? = (a + b)(a – b) + 2b².]
(d) gcd(a + b, a² – ab + b?) =1 or 3.
[Hint: a? - ab +b2 = (a + b)² - 3ab.]
5. For n > 1, and positive integers a, b, show the following:
(a) If gcd(a, b) = 1, then gcd(a" , b") = 1.
[Hint: See Problem 20(a), Section 2.2.]
(b) The relation a" | b" implies that a | b.
[Hint: Put d = gcd(a, b) and write a = rd,b = sd, where gcd(r, s) = 1. By part (a),
gcd(r", s") = 1. Show that r = 1, whence a = d.]
6. Prove that if gcd(a, b) = 1, then gcd(a + b, ab) = 1.
7. For nonzero integers a and b, verify that the following conditions are equivalent:
(a) a|b.
(b) gcd(a, b) =la|.
(c) Icm(a, b) = |b|.
8. Find lcm(143, 227), lcm(306, 657), and lcm(272, 1479).
9. Prove that the greatest common divisor of two positive integers divides their least common
multiple.
10. Given nonzero integers a and b, establish the following facts concerning lcm(a, b):
(a) gcd(a, b) = lcm(a, b) if and only if a = ±b.
(b) If k > 0, then lcm(ka, kb) = k lcm(a, b).
(c) If m is any common multiple of a and b, then lcm(a, b)|m.
[Hint: Put t =
0 <r <t. Show that r is a common multiple of a and b.]
Icm(a, b) and use the Division Algorithm to write m = qt +r, where
11. Let a, b, c be integers, no two of which are zero, and d = gcd(a, b, c). Show that
d = gcd(gcd(a, b), c) = gcd(a, gcd(b, c)) = gcd(gcd(a, c), b)
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