# Find number of soldiers in a field given the following conditions? [closed]

I could not approach any logic To find number of soldiers in a field.Can anyone guide to solve the problem?

## closed as off-topic by Servaes, user223391, Fabian, Harish Chandra Rajpoot, Claude LeiboviciNov 1 '15 at 8:58

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• Just check each of the five possibilities to see if the remainders when dividing by 8, 15, 20, 9 and 13 are as specified. (Since you only have a few options to choose between, you don't even need to assume the remaining soldiers are Chinese. In fact, just checking for $N\equiv 1\pmod{20}$, which you can do in your head, eliminates all options except one). – Henning Makholm Nov 1 '15 at 3:02
• Using process of elimination, 121 is the only number that has a remainder of 4 when divided by 9. – Taylor Nov 1 '15 at 3:03

\begin{align*} \left\{\begin{array}{ll} x&\equiv 1 \pmod{8=2^3}\\ x&\equiv 1 \pmod{15=3\times 5}\\ x&\equiv 1 \pmod{20=2^2\times 5}\\ x&\equiv 4 \pmod{9=3^2}\\ x&\equiv 4 \pmod{13} \end{array}\right\}\iff \left\{\begin{array}{ll} x&\equiv 1 \pmod{2^3}\\ x&\equiv 1 \pmod{5}\\ x&\equiv 4 \pmod{9}\\ x&\equiv 4\pmod{13} \end{array}\right\}\\ \iff x\equiv 121 \pmod{2^3\times 5\times 9\times 13 =4680} \end{align*} by Chinese Remainder Theorem.
• Once you use knowledge about the options, $x\equiv1\pmod{20}$ alone is enough to eliminate all the other options. So if you drag in Chinese remainders it can only be because you want to ignore the given options -- and then you ought to use all the information and reach the complete solution $x=121+4680k$ instead. – Henning Makholm Nov 1 '15 at 3:14