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vfiekz [6]
2 years ago
15

Find the smallest 4 digit number such that when divided by 35, 42 or 63 remainder is always 5

Mathematics
1 answer:
alex41 [277]2 years ago
5 0

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.

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\text {Hello! Let's solve this Problem!}

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\text {If the number is 5 or more you raise the score. Meaning you add 1 to the}\text {thousandth place. If the number is 4 or lower you keep the number the same.}

\text {6 is greater than 5. So this means you add 1 to 8 and it becomes 9. Every}\text {number behind it becomes 0.}

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3 years ago
Pls help …. Need it immediately ….
Leno4ka [110]

The answer is 10.73 years.

Here, the formula we can use is :

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Substitute the values mentioned in the question.

10.54 = (10.35 + Girls)/2

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3 0
2 years ago
What is the standard form of the equation y=-5/2x-3 ?
butalik [34]

The standard form of an equation is the form where that equation has no fractions and is written in the form ax + by = c.


To get rid of the fractions in y = -5/2 x - 3, we need to multiply both sides by 2. That makes the equation become:


2y = -5x - 6


Now we need to get it into the form ax + by = c. We can do this by adding 5x to each side.


5x + 2y = -6 is our answer

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