Mathematics: Find the smallest possible value of a whole number y if it leaves a remainder of 3 when divided by 5,6 or 9.

Find the smallest possible value of a whole number

Ответ:

levilugar

10.03.2022

Mathematics

We want to find such that

$\begin{cases}y\equiv3\pmod5\\y\equiv3\pmod6\\y\equiv3\pmod9\end{cases}$

6 and 9 are not coprime, so we split the moduli according to $6=2\cdot3$ and 9=3^2 to get the system

$\begin{cases}y\equiv3\equiv1\pmod2\\y\equiv3\equiv0\pmod3\\y\equiv3\pmod5\end{cases}$

If we take

$y=1\cdot3\cdot5+0\cdot2\cdot5+3\cdot2\cdot3$

we can see that

taken mod 2, the last two terms vanish and we're left with $15\equiv1\pmod2$ ;taken mod 3, the first and last terms vanish, and the remaining term is $0\pmod3$ ;taken mod 5, the first two terms vanish, and we're left with $18\equiv3\pmod5$

By the Chinese remainder theorem, we've found that any satisfying

$y\equiv15+0+18\equiv33\pmod{2\cdot3\cdot5}\equiv3\pmod30$

will satisfy each congruence above, and that any solution of the form y=3+30n for any integer will work.

The smallest possible value of these occurs for n=0 , so that 3 is the least positive solution.

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Ответ:

madmar

09.11.2021

Mathematics

No, because two points with the same x-value have different y values.

Step-by-step explanation:

The two points in the middle of the graph are (4, 2) and (4, 10). As can be seen, the x-value for both of the points are 4. Functions are when you multiply the x value by the slope (the same number every time) to get the answer for y. You cannot get two different answers by mutliplying the same numbers together.

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Find the smallest possible value of a whole number 'y' if it leaves a remainder of 3 when divided by 5,6 or 9.

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