Homework 4 Solutions
4 thg 1 2016 4.1.37 [2 points] Find counterexamples to the following claims: 1. If ac ? bc (mod m) and m ? 2 then a ...
3. Applications of Number Theory 3.1. Representation of Integers
Proof. Suppose GCD(c m) = 1 and ac ? bc(mod m). (We may assume m > 1 so that c = 0.) Then ac ? bc = c(a ? b) = km for some integer k. This implies c
c (False). (7) If a ? b(mod m)
then ac ? bd(mod m). (True). (8) If ac ? bc(mod m)
Number Theory - Modular arithmetic and GCD
22 thg 9 2013 If ac ? bc (mod m)
Number Theory and Graph Theory Chapter 2 Prime numbers and
Hence a+c ? b+c (mod m) and ac ? bc (mod m). Page 4. 4. Proof of Part 6: We prove an ? bn (mod m)
Problem Set 4 Solutions
22 thg 2 2005 Prove all of the following statements except for the two that are ... (f) ac ? bc (mod n) implies a ? b (mod n) provided c ? 0 (mod n).
Number Theory
If a ? b (mod m) then ac ? bc (mod m) for any integer c. If a ? b (mod m) and c ? d (mod m)
LECTURE 3: CONGRUENCES 1. Basic properties of congruences
(v) If a ? b (mod m) and c > 0 then ac ? bc (mod mc). Proof. Verification of these properties is straightforward. For instance
bc for all integers c and By Fact M2 (iii)
then ac ? bc(mod m.) You should be.
MATH 115 SUMMER 2012 LECTURE 4 THURSDAY
https://math.berkeley.edu/~mcivor/math115su12/lectures/lecture4.pdf
Everything You Need to Know About Modular Arithmetic
We have the following rules for modular arithmetic: Sum rule: IFa?b(modm) THENa+c?b+c(modm) (3) Multiplication Rule: IFa?b(modm) and if c?d(modm) THENac?bd(modm) (4) De?nitionAn inverse toamodulomis a integerbsuch that ab?1(modm) (5) By de?nition (1) this means that ab?1 =k· mfor some integer k
How to prove this: If a?b (mod m) and c is any integer then - Quora
Proposition 1 For any integers abcm such that m > 1 and gcd(cm) = 1 if ac ? bc (mod m) then a ? b (mod m) Proof : If m[(a?b)c] and gcd(cm) = 1 by a property we proved in class we have m(a?b) Division of a real number x by another real y 6= 0 is the same as multiplying x by 1/y (also written as y?1) and that y?1 is also
Modular Arithmetic - Department of Computer Science
If a ? b (mod m) and c ? d (mod m) then – a + c ? b + d (mod m) – ac ? bd (mod m) E g 11 ? 1 (mod 10) ? 11999 ? 1999 ? 1 (mod 10) 9 ? –1 (mod 10) ? 9999 ? (–1)999 (mod 10) 7999 ? 49499 7 ? (–1)499 7 ? –7 ? 3 (mod 10)
Congruences (Part 1) - University of Toronto Department of
5 ? 9 (mod 4) 2) If a ? b (mod m) & c ? d (mod n) then ac ? bd (mod m) Proof: a = b + qm c = d + rm some rq ac – bd = (b + qm) ( d + rm) – bd = bd + brm + qmd +qrm2 – bd = m (br + qd + qrm) Divisible by m so ac ? bd (mod m) Corollary: If a ? b (mod m) then a2 ? b2 (mod m) Proof: Case where a = c & b = d 3) If a ? b
Theorem Definition Example
Suppose GCD(c;m) = 1 and ac bc(mod m) (We may assume m>1 so that c6= 0 ) Then ac bc= c(a b) = km for some integer k This implies cjkm: Since GCD(c;m) = 1 Lemma 2 from Module 5 2 asserts that if cjkm then cjk: Write k= cdfor some integer d and substitute for kin the equation above: c(a b) = km= (cd)m= c(dm): 3 APPLICATIONS OF NUMBER THEORY 165
Searches related to if ac à bc mod m and c filetype:pdf
Supposea=b (modm) andb=c (modm) Then thereare integersjandksuch that a?b=jm b?c=km Add the two equations: a?c= (j+k)m This implies thata=c (modm) Theorem Supposea=b (modm) andc=d (modm) Then: a+c=b+d (modm) ac=bd (modm) Note that you can use the second property and induction to show that ifa=b (modm) then an=bn (modm) for all n?1
Is a-c(b-c)(mod m)?
- Is a-c? (b-c) (mod m)? Yes, this follows directly from the fact that it holds for addition, given that . However it does not hold for division - if and an integer with and both integers, it does not necessarily follow that . For example , but (. Furthermore, I am of the opinion that Quora should re-introduce the suggest edits feature.
Is a-c(b-c)(mod m) the same for subtraction?
- We know that, if a?b (mod m) and c is an integer, then a+c? (b+c) (mod m). It is the same with multiplication too but is it the same for subtraction also? Is a-c? (b-c) (mod m)? Yes, this follows directly from the fact that it holds for addition, given that .
What does a b (mod m) mean?
- 1) If a ? b (mod m) then (a+c) ? (b+d) (mod m) & c ? (mod m) Proof: a ? b (mod m) means a – b = mq, some q ? Z Also c – d = mr some r ? Z.
What does a is congruent to b modulo m mean?
- "a is congruent to b modulo m" means m|(a-b). Equivalently, a & b leave the same remainder by division by m (for a,b?0) 1) If a ? b (mod m) then (a+c) ? (b+d) (mod m) & c ? (mod m) Proof: a ? b (mod m) means a – b = mq, some q ? Z Also c – d = mr some r ? Z.
