1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
valentina_108 [34]
3 years ago
5

Solve the following simultaneous linear congruences.

Mathematics
1 answer:
Anastaziya [24]3 years ago
5 0

a. The moduli are coprime, so you can apply the Chinese remainder theorem directly. Let

x=4\cdot5+3\cdot5+3\cdot4

  • Taken mod 3, the last two terms vanish, and 20\equiv2\pmod3 so we need to multiply by the inverse of 2 modulo 3 to end up with a remainder of 1. Since 2\cdot2\equiv4\equiv1\pmod3, we multiply the first term by 2.

x=4\cdot5\cdot2+3\cdot5+3\cdot4

  • Taken mod 4, the first and last terms vanish, and 15\equiv3\pmod4. Multiply by the inverse of 3 modulo 4 (which is 3 because 3\cdot3\equiv9\equiv1\pmod4), then by 2 to ensure the proper remainder is left.

x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4

  • Taken mod 5, the first two terms vanish, and 12\equiv2\pmod5. Multiply by the inverse of 2 modulo 5 (3, since 3\cdot2\equiv6\equiv1\pmod5) and again by 3.

x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4\cdot3\cdot3

\implies x=238

By the CRT, we have

x\equiv238\pmod{3\cdot4\cdot5}\implies x\equiv-2\pmod{60}\implies\boxed{x\equiv58\pmod{60}}

i.e. any number 58+60n (where n is an integer) satisifes the system.

b. The moduli are not coprime, so we need to check for possible contradictions. If x\equiv a\pmod m and x\equiv b\pmod n, then we need to have a\equiv b\pmod{\mathrm{gcd}(m,n)}. This basically amounts to checking that if x\equiv a\pmod m, then we should also have x\equiv a\pmod{\text{any divisor of }m}.

x\equiv4\pmod{10}\implies\begin{cases}x\equiv4\equiv0\pmod2\\x\equiv4\pmod5\end{cases}

x\equiv8\pmod{12}\implies\begin{cases}x\equiv0\pmod2\\x\equiv2\pmod3\end{cases}

x\equiv6\pmod{18}\implies\begin{cases}x\equiv0\pmod2\\x\equiv0\pmod3\end{cases}

The last congruence conflicts with the previous one modulo 3, so there is no solution to this system.

You might be interested in
What's the answer to 1&2 plzz
Viktor [21]
For number one the answer is A 2,880
for number two the answer is C 75% <--- i guessed 
hope this helped
3 0
3 years ago
According to some students, What's is the true purpose of homework? 19 letters page 156 Algebra 1 worksheet
Bingel [31]

(wouldn't this be English?)
The true purpose of homework to me is to learn more and to help you remember what you have learned.
4 0
3 years ago
Multiply. <br> (5x - 2)(8x + 3) <br> Enter your answer, in standard form, in the box.
ziro4ka [17]
Standard form:

40x^2−x−6
8 0
3 years ago
3<br> 4<br> ÷ <br> 5<br> 12<br> = <br> 3 answer quick for brainly<br> 4<br> ×
tresset_1 [31]

Answer:

12

Step-by-step explanation:

3 0
3 years ago
Can someone help me find x please.
valina [46]

Answer:

x =3 sqrt(3)

Step-by-step explanation:

cos theta = adj/ hyp

cos 60 = x/ 6 sqrt(3)

Multiply each side by 6 sqrt(3)

6 sqrt(3) cos 60 = x/6 sqrt(3)/ 6 sqrt(3)

x =cos 60 * 6 sqrt(3)

x =3 sqrt(3)

4 0
3 years ago
Other questions:
  • You are charged 6% sales tax. You purchase a new bicycle and pay $27 in sales tax. What is the purchase price of the bicycle?
    11·1 answer
  • Solve the system of equation by elimination. <br> 6x-3y=-21<br> 4x=3y-9
    8·1 answer
  • Evaluate the expression. 42+5⋅32−42÷23 CLEAR CHECK 12.625 222 45 59
    10·1 answer
  • In a Gallup telephone survey conducted on April 9-10, 2013, the person being interviewed was asked if they would vote for a law
    9·1 answer
  • Determine whether the value given below is from a discrete or continuous data set. In a test of a method of gender selection, 72
    15·1 answer
  • Choose the ratios for sin A and cos A.
    11·1 answer
  • F(x) = x^2. what us g(x)?​
    12·1 answer
  • Which interval is the solution set to 0.35x – 4.8 &lt; 5.2 – 0.9x? (–∞, –8) (–∞, 8) (–8, ∞) (8, ∞)
    14·2 answers
  • Circle the reason for each of the following manipulations used to simplify the product (8x^2) (3x^7) ?
    12·2 answers
  • 4a+b−5<br><br> A 2<br> B 6<br> C 7<br> D 14
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!