Explanation As per the division algorithm, let a , b ∈ Z , b ≠ 0 . Then there exists unique integers q and r , with 0 ≤ r < | b | , such that a = q b + r . Also, if a , b > 0 and q , r are chosen to satisfy the requirements of the Division Algorithm, then q = { ⌊ a b ⌋ when b > 0 ⌈ a b ⌉ when b < 0 . We are given that a = 12 , 345 and b = − 39 . As, b < 0 , q = ⌈ a b ⌉ . q = ⌈ 12345 − 39 ⌉ = ⌈ − 316 .538 ⌉ = − 316 Also, 12 , 345 = − 316 × − 39 + 21 Thus, the quotient q is − 316 and the remainder r is 21. (b) We are given that a = − 27 , 361 and b = − 977 . As, b < 0 , q = ⌈ a b ⌉ . q = ⌈ − 27361 − 977 ⌉ = ⌈ 28 .005 ⌉ = 29 Also, − 27 , 361 = 29 × − 977 + 972 Thus, the quotient q is 29 and the remainder r is 972. (c) We are given that a = − 102 , 497 and b = − 1473 . As, b < 0 , q = ⌈ a b ⌉ . q = ⌈ − 102497 − 1473 ⌉ = ⌈ 69 .583 ⌉ = 70 Also, − 102 , 497 = 70 × − 1473 + 613 Thus, the quotient q is 70 and the remainder r is 613. We are given that a = 98 , 764 and b = 4789

3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Chapter 4.1, Problem 6E

To determine

The integers q and r, with 0≤r<|b|, such that a=bq+r in each of the following cases.

(a) a=12,345;b=−39

(b) a=−27,361;b=−977

(c) a=−102,497;b=−1473

(d) a=98,764;b=4789

(e) a=−41,391;b=−755

(f) a=555,555,123;b=111,111,111

(g) a=81,538,416,000;b=38,754

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Chapter 4.1, Problem 5E

Chapter 4.1, Problem 7E

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Discrete Mathematics with Graph Theory

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Solution Summary: The author explains how to determine the integers q and r, using the division algorithm.

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The integers q and r, with 0≤r<|b|, such that a=bq+r in each of the following cases. (a) a=12,345;b=−39 (b) a=−27,361;b=−977 (c) a=−102,497;b=−1473 (d) a=98,764;b=4789 (e) a=−41,391;b=−755 (f) a=555,555,123;b=111,111,111 (g) a=81,538,416,000;b=38,754

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Chapter 4 Solutions

The integers q and r, with 0≤r<|b|, such that a=bq+r in each of the following cases.

(a) a=12,345;b=−39

(b) a=−27,361;b=−977

(c) a=−102,497;b=−1473

(d) a=98,764;b=4789

(e) a=−41,391;b=−755

(f) a=555,555,123;b=111,111,111

(g) a=81,538,416,000;b=38,754