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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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Solution:

The total number of possiblities rolling a number cube with faces numbered 1 to 6 is;

And the total number of possibilities spining the spinner is;

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Hence, the unique combination is;

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Step-by-step explanation:

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There is a list of states with lower sales tax rates:

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3 years ago
What is the linear function equation that best fits the data set?
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A

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Which equation represents a line that passes through (5, 1) and has a slope of 1/2?
lesya692 [45]

Answer:

C

Step-by-step explanation:

The equation of a line in point- slope form is

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where m is the slope and (a, b) a point on the line

here m = \frac{1}{2} and (a, b) = (5, 1), so

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