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Ksju [112]
3 years ago
6

Find the smallest positive $n$ such that \begin{align*} n &\equiv 3 \pmod{4}, \\ n &\equiv 2 \pmod{5}, \\ n &\equiv

6 \pmod{7}. \end{align*}
Mathematics
1 answer:
Alex777 [14]3 years ago
3 0

4, 5, and 7 are mutually coprime, so you can use the Chinese remainder theorem right away.

We construct a number x such that taking it mod 4, 5, and 7 leaves the desired remainders:

x=3\cdot5\cdot7+4\cdot2\cdot7+4\cdot5\cdot6

  • Taken mod 4, the last two terms vanish and we have

x\equiv3\cdot5\cdot7\equiv105\equiv1\pmod4

so we multiply the first term by 3.

  • Taken mod 5, the first and last terms vanish and we have

x\equiv4\cdot2\cdot7\equiv51\equiv1\pmod5

so we multiply the second term by 2.

  • Taken mod 7, the first two terms vanish and we have

x\equiv4\cdot5\cdot6\equiv120\equiv1\pmod7

so we multiply the last term by 7.

Now,

x=3^2\cdot5\cdot7+4\cdot2^2\cdot7+4\cdot5\cdot6^2=1147

By the CRT, the system of congruences has a general solution

n\equiv1147\pmod{4\cdot5\cdot7}\implies\boxed{n\equiv27\pmod{140}}

or all integers 27+140k, k\in\mathbb Z, the least (and positive) of which is 27.

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Wen is factoring the polynomial, which has four terms. 6x3 – 12x2 7x – 14 6x2 (x – 2) 7(x – 2) Which is the completely factored
-Dominant- [34]

The completely factored form of the Wen's polynomial, which has the four terms initial, is,

(6x^2+7)(x-2)

<h3>What is the factor of polynomial?</h3>

The factor of a polynomial is the terms in linear form, which are when multiplied together, give the original polynomial equation as result.

Wen is factoring the polynomial, which has four terms.

6x^3 - 12x^2+ 7x - 14 '

Take out the greatest common factor from the equation and make separate groups as,

6x^2(x - 2)+ 7(x - 2)\\(x-2)(6x^2+7)

Rearrange the above equation as,

(6x^2+7)(x-2)

Thus, the completely factored form of the Wen's polynomial, which has the four terms initial, is,

(6x^2+7)(x-2)

Learn more about factor of polynomial here;

brainly.com/question/24380382

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2 years ago
Asking again because last time I didnt get an actual answer. What is the surface area of this composite figure?
Katyanochek1 [597]

Answer:

644 cm^2

Step-by-step explanation:

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10*12-3*3=111

428+111=539

8*3*4+3*3=96+9=105

539+105=644

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3 years ago
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