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-- Apr 4 In-Class Exercise Thread

If ax = b(mod n) either has d distinct solutions modulo n or no solutions: where d = gcd(a,n)

For 5x=3mod15: d = gcd(5,15) = 5 for the solution to be 5 => gcd(a,n)|b but 5 does not divide 3. So, there is no solution.

For 10x=5mod15: d = gcd(10,15) = 5 for the solution to be 5 => gcd(a,n)|b 5 divides 10. Hence the following are the solutions: solutions are 2,5,8,11,14

(Edited: 2018-04-04)

If ax = b(mod n) either has d distinct solutions modulo n or no solutions: where d = gcd(a,n) For 5x=3mod15: d = gcd(5,15) = 5 for the solution to be 5 => gcd(a,n)|b but 5 does not divide 3. So, there is no solution. For 10x=5mod15: d = gcd(10,15) = 5 for the solution to be 5 => gcd(a,n)|b 5 divides 10. Hence the following are the solutions: solutions are 2,5,8,11,14

-- Apr 4 In-Class Exercise Thread

 

Q1). 5x = 3 mod 15
	GCD(5,15) = 5. But 5 is not dividable by 3. So there are no solutions to this equation.
Q2). 10x = 5 mod 15
	 GCD(10,15)= 5. 5 is divisible by 5. so, this equation has 5 solutions.
Q3). Solutions are 2,5,8,11,14

<nowiki> Q1). 5x = 3 mod 15 GCD(5,15) = 5. But 5 is not dividable by 3. So there are no solutions to this equation. Q2). 10x = 5 mod 15 GCD(10,15)= 5. 5 is divisible by 5. so, this equation has 5 solutions. Q3). Solutions are 2,5,8,11,14 </nowiki>

-- Apr 4 In-Class Exercise Thread

Name : Kunal Deshmukh There will be no solution satisfying criteria in Corollary. There is a solution for 2nd equation as 5|gcd(10,15) . no of solutions for this eqn are 5.

<nowiki> Name : Kunal Deshmukh There will be no solution satisfying criteria in Corollary. There is a solution for 2nd equation as 5|gcd(10,15) . no of solutions for this eqn are 5. </nowiki>

-- Apr 4 In-Class Exercise Thread

 1. 5x = 3mod15
    a = 5
    b = 3
    n = 15
    d = gcd(5,15) = 5
    Since b does not divide d i.e 3 does not divide 5
    This equation has no solution
 2. 10x = 5mod15
    a = 10
    b = 5
    n = 15
    d = gcd(5,15) = 5
    Since b divided d i.e 5 divides 5
    We have b solutions = 5
    10*2 mod 15 = 5 
    10*3 mod 15 = 0
    so 2,(2+3)=5, (5+3)=8,(8+3)=11,(11+3)=14 are the solutions

1. 5x = 3mod15 a = 5 b = 3 n = 15 d = gcd(5,15) = 5 Since b does not divide d i.e 3 does not divide 5 This equation has no solution 2. 10x = 5mod15 a = 10 b = 5 n = 15 d = gcd(5,15) = 5 Since b divided d i.e 5 divides 5 We have b solutions = 5 10*2 mod 15 = 5 10*3 mod 15 = 0 so 2,(2+3)=5, (5+3)=8,(8+3)=11,(11+3)=14 are the solutions

-- Apr 4 In-Class Exercise Thread

 1. No solutions since gcd(5, 15) = 5 does not divide 3
 2. 5 solutions -> 2, 5, 8, 11, 15

(Edited: 2018-04-08)

1. No solutions since gcd(5, 15) = 5 does not divide 3 2. 5 solutions -> 2, 5, 8, 11, 15