Unformatted text preview: 46 10. 11.
12. 13. 14.
15. 16. Chapter 2 Congruences (d) Formulate a conjecture based on the (rather limited) numerical
evidence above. (a) Formulate a conjecture as to why Z4 and ZS in Example 4 are different
structures algebraically. [Hintz Examine the nonzero entries of the
corresponding multiplication tables. (A brief discussion of rings and
ﬁelds can be found in Appendix C.)] (b) Construct addition and multiplication tables modulo 6. Is Z6 a ﬁeld? (c) Construct addition and multiplication tables modulo 7. Is Z7 3 ﬁeld? (11) On the basis of work in the text. your results in parts (a) and (b)
above, and further experimentation. complete the following
conjecture: Conjecture: Z,, is a ﬁeld if and only if n is
(Hint: Experiment further!) Let a, b EZ such that a E b mod m. If n is a positive integer such that n j m, prove that a E b mod n. Let a, b EZ such that a E b mod m. If c is a positive integer. prove that ca E cb mod cm. Let a, b 62 such that a E b mod m. If d is a positive integer with d j a, d I b, and d j m, prove that :1, E gmod’ﬁ'. Let a, b e Z such that a E b mod m. Prove that (a, m) E (b, m). Let a, b, c, (162 such that a E b modm and c E dmod m. Prove or disprove the following statements. (a) (a E c) E (b E d) modm (from which it would follow that the sub-
traction of equivalence classes under congruence modulo m is well
deﬁned). (11) lfcja anddlb, then f: E Zmodm. (a) Let a be an even integer. Prove that a2 E 0 mod 4. (b) Let a be an odd integer. Prove that a2 E 1 mod 8. Deduce that
a2 E 1 mod 4. (c) Prove that if n is a positive integer such that n E 3mod 4. then n
cannot be written as the sum of two squares of integers. (d) Prove or disprove the converse of the statement in part (c) above. Let n be an odd integer not divisible by 3. Prove that n2 E 1 mod 24. Let a,bEZ with a E bmod m. If n is a positive integer, prove that a” E b" mod m. (Note: Exercises 16—19 below establish divisibility tests. As a global hint
for these exercises, you will ﬁnd it helpful to use Exercise 15 above as well
as the fact that any positive integer n can be written in expanded base 10 form as
n E a,,,10’" + a,,.,.110'"’1+~-+ a110l+ an where each a,- is one of the digits 0.1.2, . . . , 9.)
Prove that a positive integer n is divisible by 2 (respectively 5) if and only
if its units digit is divisible by 2 (respectively 5). (Hint: For divisibility by
2, note that 10 E 0 mod 2. from which Proposition 2.4 implies that
am10m+ a,,,,110"”1+~~+ a1101+ a0 E a,,.0'" + am ,10'"1 + --' + a101 + do mod 2) ...
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