not quite understanding...

Question

  2.)    When  Jason  arranges  his  toy  soldiers  in  rows  of  6,  he  has  1  left  over.    When  he   arranges  them  in  rows  of  8,  he  has  3  left  over.    When  he  arranges  them  in  rows  of   10,  he  has  5  left  over.    What is the smallest possible number of toy soldiers Jason plays with?

Answer 1

This is similar to the concept of least common multiple, but more complicated, because the number of soldiers isn't a multiple of 6, 8 or 10. The easiest way to do this is trial-and-error.

 

If there are 15 soldiers, that would make one row of 10, with 5 left over. It would make 2 rows of 6, but with 3 left over, so we know the total isn't 15. I got 15 by taking 10x1 + 5 (1 row of 10 with 5 left over). Now with 2 rows of 10 and 5 left over, that's 25 soldiers.

25 soldiers would be 4 rows of 6 with 1 left over, so this might be right.

It would be 3 rows of 8 with 1 left over, so 25 isn't it, either.

 

OK, let's try 3 rows of 10 plus 5, which is 35 soldiers.

That's 5 rows of 6, but 5 left over. Nope.

 

45 makes 7 rows of 6 with 3 left over. No.

 

55 makes 9 rows of 6 with 1 left over. OK so far.

55 makes 6 rows of 8 with 7 left over. No.

 

65? No.

75? No.

 

OK, this trial and error thing is taking too long. Let's approach it mathematically (number theory). Let's let "n" represent the number of soldiers. We know that n-5 is divisible by 10, that n-1 is divisible by 6, and that n-3 is divisible by 8. (This is because of the remainders when we divide by those numbers). Now, we can return to trial-and-error, but in a more systematic way. The number has to end in 5 (because of having 5 left over when divided by 10) and be such that the number minus 1 is divisible by 6, the number minus 3 is divisible by 8, and the number minus 5 is divisible by 10. A chart is very helpful now

 

Number      n-1          n-3          n-5

75               74 is not div by 6

85               84 not by 6

95               94 not by 6

105            104 not by 6

115            114 not by 6

125            124 not by 6

135            134 not by 6

145            144 yes   142 not by 8

155            154 not by 6

165            164 not by 6

175            174 yes   172 not by 8

185            184 no

195            194 no

205            204 yes   202 no

215            214 no

225            224 no

235            234 yes   232 yes   230 yes

 

So, he has at least 235 toy soldiers, which can make 39 rows of 6, with one left over, 29 rows of 8, with 3 left over, and 23 rows of 10, with 5 left over. There is probably a straightforward way of accomplishing this, but it is beyond my current abilities. When I was studying number theory in college, I'm sure we learned about these, but that was a while ago.