Math Is Fun Forum

TARAJS

Member

Registered: 2011-12-01

Posts: 19

Two random things

THree horses run a race. How many different ways can the three finish if ties are allowed?
I used permutation of n^r so 3^3=27 different ways for them to finish. Just want to make sure this is correct?

Also, Which of the following numbers is a perfect square? ( I need to know to figure this out without a calculator and how to explain it)

329476 (i know this is the answer) 389372    964328     326047    and 724203

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Two random things

hi TARAJS,

I've posted about your horses in the other post.  These three need a rest as they've already had a race today! 

Now the squares.

0 x 0 = 0
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
.......

9 x 9 = 81

Look at the unit digits.

Only 0, 1, 4, 5, 6, 9 ever come up for the units digit of the square.

Larger squares obey the same rule eg.  16 x 16 ends in 6; 213455 x 213455 ends in 5.

You will never get a square number ending in 2 or 3 or 7 or 8.

So look at your possible numbers.

389372    964328     326047     724203

You can reject all of these using this rule alone.

So, if you know exactly one number is the square, it has to be the first.  329476

What do you do if you don't know for certain that it is a square?

It is a contender as it ends in 6.  That doesn't prove it is a square.  26 ends in 6 and it isn't!

You can try to split the number into its prime decomposition.

What that means is you look for possible factors, divide by them and then try to split what's left into factors in the same way.

329476 is even so 2 is a factor, 329476 / 2 = 164738.

OK, the answer is still even so 2 is still a factor, 164738 / 2 = 82369

3 won't go, nor 5 but 7 will.  82369 / 7 = 11767

Try 7 again, yes, 11767 / 7 = 1681

Now it gets harder to spot factors.

To end in a 1 a factor must be  *1 or *9 where * is any possible tens digit.

So try 11, no, 19, no, 21, no (as we've already rejected 3 and 7), 29, no, 31, no, 39 no ( as 3 won't go), 41 yes.

1681 / 41 = 41.

So we end up with this product of primes:

329476 = 2 x 2 x 7 x 7 x 41 x 41 = (2 x 7 x 41) x (2 x 7 x 41)

So it is a perfect square (2 x 7 x 41) squared = 574 x 574

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

TARAJS

Member

Registered: 2011-12-01

Posts: 19

Re: Two random things

Thank you!