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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.

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Answer:

<em>The second figure ( rectangle ) has a longer length of it's diagonal comparative to the first figure ( square )</em>

Step-by-step explanation:

We can't confirm the length of these diagonals based on the appearance of the figure, so let us apply Pythagorean Theorem;

This diagonal divides each figure ( square + rectangle ) into two congruent, right angle triangles ⇒ from which we may apply Pythagorean Theorem, where the diagonal acts as the hypotenuse;

5^2 + 5^2 = x^2 ⇒ x is the length of the diagonal,

25 + 25 = x^2,

x^2 = 50,

x = √50

Now the same procedure can be applied to this other quadrilateral;

3^2 + 7^2 = x^2 ⇒ x is the length of the diagonal,

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<em>Therefore the second figure ( rectangle ) has a longer length of it's diagonal comparative to the first figure ( square )</em>

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Step-by-step explanation:

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Read 2 more answers
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