You think therefore you am.

such wise of a post.

]]>You're welcome (I think)!

]]>I might have worried all night about what that meant. I'll check the site out and see what it is about.

Bless you!

]]>AoPS is another forum. It's full name is Art of Problem Solving.

]]>You are quite welcome! Seeing patterns is often the name of the game in solving math problems.

And if you have access to a good math program you can often write a program to test for patterns

associated with a given problem.

Hi cooljackiec, Not only am I not levans, I have no clue as to what AoPS is. What is it?

And thanks bobbym for the kind comment. I do try to be polite and not hurt anyone's feelings

even if I think what they are saying is incorrect. After all, I have been wrong before. As I

recall once in 1947 I was wrong about ...

epecially noel as m a student of class 9 and thus his soln ws much more easy fr me:).]]>

I do not think so. Noel is polite.

]]>0th power of 12357 ends in 1.

1st power ends in 7. 2nd power ends in 9. 3rd power ends in 3. 4th power ends in 1

5th power ends in 7 6th power ends in 9 7th power ends in 3. 8th power ends in 1

9 ... 7 10 ... 9 11 ... 3 12 ... 1, etc.

There are only four digits that occur in the powers and these repeat in the sequence 1, 7, 9, 3, 1, ...

Divide the power (655 in this case) by four and see what the remainder is. It is three. So the 655th

power ends in the same digit that the 3rd power does, namely 3.

But this is a bit confusing since we get 3 for the remainder and for the last digit.

Try raising to the 656 power. Then dividing 656 by 4 we get a remainder of zero. So the 656

power ends in 1 same as the 0th power.

For the 654th power dividing 654 by 4 has remainder 2. So the 654th power ends in 9 the same

as the second power.

For the 653rd power dividing 653 by 4 has remainder 1 so the units digit of the 653rd power is 7 the

same as the 1st, 5th, 9th, ... powers.

Be very blesses!

]]>Welcome to the forum. It is a 3.

Answer is 3.

]]>please tell me the solution!!

]]>