A General And His Soldiers

Number Theory Level 3

In ancient China, generals counted soldiers remaining after a battle by lining them up in rows of diﬀerent lengths.

If a general had $1200$ soldiers at the start of a battle and if there were

$1)$ $3$ left over when they lined up $5$ at a time;

$2)$ $3$ left over when they lined up $6$ at a time;

$3)$ $1$ left over when they lined up $7$ at a time;

$4)$ none left over when they lined $11$ at a time,

how many soldiers survived the battle?

The answer is 1023.

2 solutions

Arpit MIshra
Apr 18, 2015

Answer- $\boxed{1023}$

int x; //Intializing x
        for(x=1200;x>=1;x--) //counting backwards
        {
            if(x%5==3&&x%6==3&&x%7==1&&x%11==0) //Eg: 3 left when lined up in 5's is comparable to modulous
            System.out.println(x);
        }

Output- 1023

Julian Poon
Aug 16, 2014

I solved this with pattern recognition. I am sure there is a more elegant way to solve this than my method.

there's nothing really more elegant than repeated applications of CRT.

mathh mathh - 6 years, 10 months ago

yup! CRT is very easy

Mohamed Moanis - 6 years, 9 months ago

Actually... I don't understand CRT. All the number theory questions that I solved and requires CRT I solve with a more inelegant way called algebra.

Julian Poon - 6 years, 8 months ago

A General And His Soldiers

The answer is 1023.

2 solutions

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