Japanese Samurai and his soldiers after a battle

A Japanese Samurai counted his soldiers remaining after a battle, by lining them in rows of different lengths, counting the number left over each time, and calculating the total from these remainders. A Samurai had 1200 soldiers at the start of a battle. After the battle, there were 3 left over when they lined up 5 at a time, 3 left over when they lined up 6 at a time, 1 left over when they lined up 7 at a time, and none left over when they lined up 11 at a time. How many soldiers were lost in the battle?

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Originally Posted by dkmathguy

A Japanese Samurai counted his soldiers remaining after a battle, by lining them in rows of different lengths, counting the number left over each time, and calculating the total from these remainders. A Samurai had 1200 soldiers at the start of a battle. After the battle, there were 3 left over when they lined up 5 at a time, 3 left over when they lined up 6 at a time, 1 left over when they lined up 7 at a time, and none left over when they lined up 11 at a time. How many soldiers were lost in the battle?

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Clearly,

So,

No. of soldiers lost in the battle

alexmahone, you are correct. Thanks, I never saw you could do negative too, lol. I did the hard way by making the -1 to a 6.

15b = -1 (mod 7)

b = -1 (mod 7)


15b = 6 (mod 7)

b = 6 (mod 7)

That is one awesome way of counting.

Can be done by simple reasoning:

a = alive: we want a / 11 = integer; so maximum a = 1199 ; then 1188, 1177, .....

we want a / 6 to have remainder of 3, so a is odd ; so a = 1199, 1177, 1155, 1133, ....

we want a / 5 to have remainder of 3, so a must end with 3 (being odd), so a = 1133, 1023, 913, ...

we want a / 7 to have remainder of 1:
1133 / 7 = 161 remainder 6
1023 / 7 = 146 remainder 1 : YA!

So deads = 1200 - 1023 = 177

Much quicker (and easier) is find minimum that accomodates 5,6,11,
which of course is 33; this number must go up by 330 (to keep accomodating 5,6,11),
so: 33, 363, 693, 1023; only 1023 has remainder of 1 if divided by 7.