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lapo4ka [179]
3 years ago
10

How to do i solve it

Mathematics
1 answer:
Leona [35]3 years ago
3 0
Use ur brain faqg lololo
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Find the smallest 4 digit number such that when divided by 35, 42 or 63 remainder is always 5
alex41 [277]

The smallest such number is 1055.

We want to find x such that

\begin{cases}x\equiv5\pmod{35}\\x\equiv5\pmod{42}\\x\equiv5\pmod{63}\end{cases}

The moduli are not coprime, so we expand the system as follows in preparation for using the Chinese remainder theorem.

x\equiv5\pmod{35}\implies\begin{cases}x\equiv5\equiv0\pmod5\\x\equiv5\pmod7\end{cases}

x\equiv5\pmod{42}\implies\begin{cases}x\equiv5\equiv1\pmod2\\x\equiv5\equiv2\pmod3\\x\equiv5\pmod7\end{cases}

x\equiv5\pmod{63}\implies\begin{cases}x\equiv5\equiv2\pmod 3\\x\equiv5\pmod7\end{cases}

Taking everything together, we end up with the system

\begin{cases}x\equiv1\pmod2\\x\equiv2\pmod3\\x\equiv0\pmod5\\x\equiv5\pmod7\end{cases}

Now the moduli are coprime and we can apply the CRT.

We start with

x=3\cdot5\cdot7+2\cdot5\cdot7+2\cdot3\cdot7+2\cdot3\cdot5

Then taken modulo 2, 3, 5, and 7, all but the first, second, third, or last (respectively) terms will vanish.

Taken modulo 2, we end up with

x\equiv3\cdot5\cdot7\equiv105\equiv1\pmod2

which means the first term is fine and doesn't require adjustment.

Taken modulo 3, we have

x\equiv2\cdot5\cdot7\equiv70\equiv1\pmod3

We want a remainder of 2, so we just need to multiply the second term by 2.

Taken modulo 5, we have

x\equiv2\cdot3\cdot7\equiv42\equiv2\pmod5

We want a remainder of 0, so we can just multiply this term by 0.

Taken modulo 7, we have

x\equiv2\cdot3\cdot5\equiv30\equiv2\pmod7

We want a remainder of 5, so we multiply by the inverse of 2 modulo 7, then by 5. Since 2\cdot4\equiv8\equiv1\pmod7, the inverse of 2 is 4.

So, we have to adjust x to

x=3\cdot5\cdot7+2^2\cdot5\cdot7+0+2^3\cdot3\cdot5^2=845

and from the CRT we find

x\equiv845\pmod2\cdot3\cdot5\cdot7\implies x\equiv5\pmod{210}

so that the general solution x=210n+5 for all integers n.

We want a 4 digit solution, so we want

210n+5\ge1000\implies210n\ge995\implies n\ge\dfrac{995}{210}\approx4.7\implies n=5

which gives x=210\cdot5+5=1055.

5 0
3 years ago
Witch one should i pick? please help me.
s2008m [1.1K]

Answer:

A

Step-by-step explanation:

The rectangle has an area of 30ft^2

Each triangle has an area of 25ft^2

25+25+30=80

6 0
2 years ago
Which of the following is equivalent to 12/20
PolarNik [594]
12/20 can be simplified to 3/5
6 0
3 years ago
Read 2 more answers
Emma ran 4 1/2 miles. For 3/4 of her run,she ran uphill. How many miles did she run uphill?
Gemiola [76]

Answer:

its c hope it helps not totally sure but yeah

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Learn with an example
lana [24]

Based on the information provided, to feed the second cat it is necessary 1 1/2 extra cans or 1.5 cans.

<h3>How much extra food is needed to feed the second cat?</h3>

This difference can be found by substracting the total cans fed with only one cat to the total fed with two cats:

  • 3 1/2 cans - 2 cans = 1 1/2 cans, which can also be expressed as 1.5 cans.

Based on this, to have a second cat implies 1.5 extra cans.

Learn more about feed in: brainly.com/question/15585750

#SPJ1

6 0
2 years ago
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