O Prove the Cance
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please help me on question 9
Transcribed Image Text:asSUCiative: ā + (b + C) = (ā + b) + č.
(c) Multiplication distributes over addition: a(b + t) = āb + a c.
(d) O is the additive identity: ā + Ō = ā.
(e) I is the multiplicative identity: ā · ī = ā.
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bom) 0
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7. Find the remainder when
*
(a) 238 + 496 – 44 is divided by 9.
(c)
421 + 366 – 14 is divided by 11. (d) 317 · 403 is divided by 9.
* (e)
152-156 is divided by 7.
(g) 182 144 is divided by 13.
AA
Joo
(b) 238+496 – 44 is divided by
(II bon
-
(f) 280 · 318 is divided by 11.
(h) 288 · 301 is divided by 13.
8. Use the methods of this section to prove that
(а)
for every integer k, 5 divides k
(b) for every integer k, 5 divides k2(k – 1).
(c) for every integer k, 5(ko – k³) = 0 or 3 (mod 7).
9. (a) Prove the Cancellation Law for Z, (Theorem 3.4.5).
* (b)
– 5k3 + 4k.
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Solve the equation 2x = 5 in Zz.
Solve the equation 2x = 5 in Z11:
(d) Solve the equation 2x = 9 in Z13.
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%3D
dxo 9Tadi (c)
(9)
10. Prove that for every natural number m, if x=y (mod m), then x" = y* (1
bas (n for all natural numbers k. 2 "loor
%3D
bus
+nS 11. Between 1970 and 2007,* commercial books were published with an I
(Huegi e (International Standard Book Number) consisting of 10 digits in the
a-bcd-efghi-j. The last digit j is a check digit, meaning that is takes o
Transcribed Image Text:upe when m is composite.
but 3 does not equal 6 in Z12.
no
The siluui
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prem 3.4.4
Let p be a prime. Whenever F ỹ = Ō in Z,, then either x= 0 or ỹ = 0
(mod p). Therefore, p divides xy. By Euclid's Lemma (Theorem 1 8 2 ty =0
clude that either p divides x or p divides y. Thus, either x = 0 (mod p Con-
(mod p). Therefore, either x =0 or y= 0.
(o bom) cTO
30+0
bas 0
A nodT (0 bo!
The cancellation law for multiplication in Z, follows directly from Theorem 3u
d.
madT o bore)
borm) 0
m 3.4.5
Cancellation Law for Z,
d.
02= Let p be a prime. If xy = xz in Z, and x#0, then y = z.
%3D
(a borm) 0
Proof. See Exercise 9(a). rIT.(o bom) A
02 -
( bom) 0
=
20 K
Chapter 6, where we look to Zm as a guiding example for arithmetic operations and
More about the algebraic properties of modular arithmetic can be found in
their properties for other algebraic systems.
A orlt ni botzil noiteoilqilum bns noiuibbe tot
esiogong od
Ani
5 3.4
brs noinibbe ol blord ozle
to loM
vi
மாக
1. Perform each of these calculations in Z,.
(a) 6+6
(mod p). p divides xy. By we con-
clude that either p divides x or p divides y. Thus, either x = 0 (mod p) or y =0
The law for in Z, from 3.44.
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