Congruence

Saturday, 9 March 2019

11:59 PM

Untitled picture.png Machine generated alternative text:
Solve the following congruences. In each question give the solution as a set, i.e.
• Ifthere are no soltions, then giveyour answer the empty set set()
• Ifthe solution is 1, then giveyour answer as set(l)
• Ifthe solutions are 1 and 2, then giveyour answer as set(1,2) .
a)
Solve 110:r
60 (mod 169).
Expected answer:
set (62)
{62}
Score: 0/1 X
unanswered

Untitled picture.png Machine generated alternative text:
b)
Solve 147T
= 36 (mod 161).
Expected answer:
set ( )
Score: 0/1 X
unanswered

Untitled picture.png Machine generated alternative text:
c)
Solve 17x=5 (mod 119).
Expected answer:
x
d)
Solve 18:r 84 (mod 160).
Expected answer:
x
Solve 161x —
— 160 (mod 167)
Expected answer:
set ( )
set (58, 138)
Score: 0/1 X
Unanswered
{58, 138}
Score: 0/1 X
nanswered
set (29)
{29}
Score: 0/1 X
Use the calculator to find GCD(110,169)Answer is 1Then find a value for x(60 + 169×𝑛)/110 -> 𝑥

𝐺𝐶𝐷﷐147, 161﷯=7
7 /| 36 so no solution
𝐺𝐶𝐷﷐17, 119﷯=17
17 /| 5 so no solution
𝐺𝐶𝐷﷐18, 160﷯=2
2 | 84 so two solutions

Brute force

Untitled picture.png Machine generated alternative text:
c)
Solve 17x=5 (mod 119).
Expected answer:
x
d)
Solve 18:r 84 (mod 160).
Expected answer:
x
Solve 161x —
— 160 (mod 167)
Expected answer:
set ( )
set (58, 138)
Score: 0/1 X
Unanswered
{58, 138}
Score: 0/1 X
nanswered
set (29)
{29}
Score: 0/1 X

Brute force
﷐84+160×6﷯/18=58
﷐84+160×15﷯/18=138

𝐺𝐶𝐷﷐18, 160﷯=1
So unique solution

Brute force
﷐160+167×27﷯/167=29

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