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N is a four-digit positive integer. Dividing N by 9 , the remainder is 5. Dividing N by 7, the remainder is 2. Dividing N by 5, the remainder is 4. What is the smallest possible value of N?

 Nov 13, 2022
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Using modular arithmetic, we can create three "equations." 

 

\(n\equiv 5 \pmod{9} \)

\(n\equiv 2 \pmod{7}\)

\(n\equiv4\pmod{5} \)

 

from the last equation, we get that \(n = 5a+4 \) for some integer a. From the middle one, we get \(n = 7b+2\) for some integer b. Set these equal to each modulo 5 to get that\(5a + 4 \equiv 7b+2 \pmod{5}\). you get that \(2b\equiv 2\pmod{5}\), so \(b \equiv 1 \pmod{5}\). So, \(b = 5c+1\) for some integer c and the bottom system simplifies to 7(5c+1) + 2 = 35c+9, so \(n\equiv 9\pmod{35} \)

 

We do the same with this and the top equation we left out before to get that \(n = 315e + 149\) for some integer e. e=3 minimizes the four digit integer, and we get n = 1094. 

 

 

hope this helped cool

 Nov 13, 2022

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