2

Find the smallest positive integer that satisfies both. x ≡ 4 (mod 9) and x ≡ 7 (mod 8) Explain how you calculated this answer.

I am taking a math for teachers course in university, so I'm worried about guessing too much in fear that I will teach myself incorrectly. Our textbook (Pearson Custom Math Text) is absolutely terrible! It skips briefly over certain elements.

If I had to guess how to do this question, it would be to convert them to x-4 and x-7 respectively then use multiples of 9 and 8 to see if I could come up with a positive integer which left the respective remainders... Is there an easier way to solve this? Am I doing it correctly?

Shayna
  • 137
  • Welcome to Mathematics.SE. I bring to your attention that Mathematics.SE is not a do-my-homework service. You should provide context, and show your attempts at the problem. A more detailed guide can be found here. Thanks in advance for your anticipated cooperation. – Lord_Farin May 21 '13 at 09:14
  • I am taking a math for teachers course in university, so I'm worried about guessing too much in fear that I will teach myself incorrectly. Our textbook (Pearson Custom Math Text) is absolutely terrible! It skips briefly over certain elements. If I had to guess how to do this question, it would be to convert them to x-4 and x-7 respectively then use multiples of 9 and 8 to see if I could come up with a positive integer which left the respective remainders... Is there an easier way to solve this? Am I doing it correctly? – Shayna May 21 '13 at 09:33
  • 1
    Thank you for responding; to make this information more visible, please [edit] it into the question body. As for the approach: It's a little sketchy, so I can't assess if you're really correct, but it can be turned into a viable strategy. An effective method can be distilled from this answer by Arturo Magidin. – Lord_Farin May 21 '13 at 09:40
  • Thank-you very much! That answer did definitely help. I'm still not totally clear on how to PROPERLY use the Chinese Remainder Theorem, but I was able to show my work for the marks... x=4+9a=7+8a 9+4=13,18+2=22,27+4=31 & 8+7=15,16+7=23,24+7=31. 31-4=27/9=3 & 31-7=24/8=3 So the answer is 31. Is there an easier way to explain the Chinese Remainder Theorem? There is nothing like it in our textbook, but I feel it may come in handy when teaching students math! – Shayna May 21 '13 at 09:55

2 Answers2

1

\begin{align} x &\equiv 4 \pmod 9 \\ x &\equiv 7 \pmod 8 \end{align}

This is the most straightforward way to a solution that I know of. The first congruence can be rewritten as $$x = 9t + 4$$ for some integer $t$. Substitute that into the second equivalence and you get

\begin{align} 9t + 4 &\equiv 7 \pmod 8 \\ 9t &\equiv 3 \pmod 8 \\ t &\equiv 3 \pmod 8 \end{align}

Whence $$t = 8n+3$$

It follows that \begin{align} x &= 9(8n+3)+4 \\ x &= 72n + 31 \end{align}

Which we write as $$ x \equiv 31 \pmod{72}$$

Steven Alexis Gregory
  • 26,769
1

Apply the Chinese Remainder Theorem (the following is just applying the proof of the theorem to this case):

$$1=1\cdot 9+(-1)\cdot 8\implies x:=(4)(-1)\cdot8+7\cdot 1\cdot9=31$$

DonAntonio
  • 211,718
  • 17
  • 136
  • 287