SOLUTION: A group of soldiers consists of less than 10000 individuals. When the commander tries to line up his soldiers in 70 equal rows, he finds that 55 soldiers were left over, when he tr

Question 536865: A group of soldiers consists of less than 10000 individuals. When the commander tries to line up his soldiers in 70 equal rows, he finds that 55 soldiers were left over, when he tries to form 65 equal rows, he finds that 50 soldiers were left over, and when he tries to form 60 equal rows, he finds that 45 soldiers were left over. Is it possible for this commander to line up his soldiers in 55 equal rows? Why or why not?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
If only that poor commander had 15 more soldiers, he would have choices. He could make 70 equal rows, or 65 equal rows, or 60 equal rows, without any soldiers left over. Do you see why? In each case, the soldiers left over plus 15 more soldiers would equal the number of rows, so you could add one soldier to each row and have no leftovers.
So, if the number of soldiers was ,
is a multiple of 60, 65, and 70.

The smallest common multiple (least common multiple, or LCM) is

That's the only choice, because the next common multiple would be twice that, and it would mean more than 10000 soldiers. So
---> which is a multiple of 5 and a multiple of 11, and therefore a multiple of 55.
You know why I see at first sight that it's a multiple of 5. To see if it's a multiple of 11, add up two groups of every other digit (ones plus hundreds and tens plus thousands). If the difference between the sums is zero or multiple of 11, the number is a multiple of 11. And if a number is a multiple of two different prime factors, it is a multiple of their product.
So the commander has 5445 soldiers and can line them up in 55 equal rows.
RELATED QUESTIONS
when a regiment is arranged in a perfect square,the number of soldiers in the front row... (answered by Edwin McCravy)

324 soldiers queued up such that the number of queues is equal to the number soldiers in (answered by Fombitz)

How many soldiers are there in a group of 27 sailors and soldiers if there are four... (answered by ewatrrr)

2017 soldiers were lined up in a file. The commander ordered them to number off... (answered by ikleyn,Alan3354)

For the Republic Day parade, 3000 soldiers were to be arranged in the form of a square.... (answered by ankor@dixie-net.com)

A battalion of soldiers wanted to have a parade forming a square design with 1536... (answered by Edwin McCravy)

In a ship of 30 people, the sailors are 2/3 of the soldiers. find the number of the... (answered by josgarithmetic)

In a ship of 30 people, the sailors are 2/3 of the soldiers. find the number of the... (answered by josgarithmetic,greenestamps)

In a ship of 30 people the sailors are 2/3 of the soldiers Find out the number of... (answered by Theo)