The Ants Go Marching

let the number of ants be x.from the given problem we can say that x-1 is a multiple of 2,x-2 is a multiple of 3,x-3 is a multiple of 4,.and so on upto x-9 is a multiple of 10.the same can be written as x-1=2a , x-2=3b , x-3=4c,............and so on upto x-9=10i....Now add their respective factor for each equality on both sides i.e...,2,3,4,5,6,7,8,9,10. we get x+1=2a+2=2(a+1),,,,,x+1=3b+3=3(b+1)..,x+1=4c+4=4(c+1) and so on upto x+1=10i+10=10(i+1)....from above we can see that x+1 is a multiple of 2,3,4,5,6,7,8,9,10.So we have x+1=LCM of (2,3,4,5,6,7,8,9,10) * k=(2520 k).therefore x=(2520 k)-1...for least value we have k=1.this implies x=(2520*1)-1=2519.............:)

L.C.M. of 1 to 10 is 2520 which means the least number divisible by 1 to 10 is 2520. Now in the problem the the ants get reduced one by one in every transformation of rows till last when 9 ants have left by some reasons and the left out ants are in 10 rows . Hence the left out ants in the last transformation must be divisible by 10 which is only possible when they start with 2519 ants.

What everyone else has written is correct, but I think it can all be done a bit more elegantly. Notice that all of the division relations can instead be written as $n\equiv-1 \mod i$ for $i=2$ to $10$ . By the Chinese Remainder Theorem, we actually only care about $i=5,7,8$ and $9$ . As others have written, the least common multiple of these numbers is $2520$ . Thus $n\equiv-1\mod2520$ , for which the smallest positive solution is $n=2519$ .

LCM of 1,2,3....10 =2520 2520-1=2519 gives you required soln

To simplify things, let's put them in symbols.

let x = min. # of ants needed.

2 | x - 1

3 | x - 2

4 | x - 3

...

10 | x - 9

Let's pretend that we only need up to 3 rows in the problem. Then, we only use up to "3 | x - 2". By Trial And Error, we find that x = 5 .

Now we add the requirement of a fourth row. We include the third statement. By a little more Trial And Error, x = 11 .

Do you notice anything? The value of x is 1 less than the LCM of the numbers from 2 up to the maximum row number.

So, [2, 3, 4, 5, 6, 7, 8, 9, 10] = 2520. 2520 - 1 = $\boxed{2519}$

:33

that song sounds good....