solve One finds in the literature the assertion that Chinese generals used to co
ID: 2971225 • Letter: S
Question
solve
One finds in the literature the assertion that Chinese generals used to count the number of their soldiers using the Chinese remainder theorem. They gave military commands to line up in several squares: Line up 7 by 7. Line up 11 by 11, and each time asked the remaining soldiers to report back. The general needs only to count these remaining soldiers. Suppose that after lining up in 7 by 7 squares, there are 10 soldiers remaining, after lining up 11 by 11, there are 8 soldiers remaining. Set up the simultaneous congruences to determine the number of soldiers. Solve these congruences and determine the number of soldiers.Explanation / Answer
49k + 10 = 121m +8
=>49k+2 = 121m => m =0.405m +2/121
=>(m > .4k)
=>24(2k-5m) + (k-m) +2 =0
=>(k-m) = 24(5m-2k) -2
As (5m-2k) > 0
=>(k-m) is of form '24x-2'
=> k-m Takes values 22,46, 70, 94, 118 ,142 ,...
Let 'k-m' = x,
=>x = 24(3m - 2(k-m)) -2
=>x=24(3m-2x)-2 =>72m =49x+2
Substituting above values of 'x',
72m = 1080, 2256, 3432, 4608, 4608, 5784, ...
m = 15, 31.33, 47.66, 64, ...
Minimum Value of m =15
=>Army strength =(121*15 + 8) =1823
(M also can take values 64,... then army strength =121*64 + 1 = 7752)
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