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1.How many of the integers between 1 and 1,000 , inclusive, can be expressed as the difference of the squares of two nonnegative integers?

2.For how many values of k is 12^12 the least common multiple of the positive integers 6^6, 8^8, and k?

3.Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms being also prime.

Enjoy!

By Loop123 (Loop123) on Friday, July 09, 2004 - 06:51 pm: Edit

Anyone up for the challenge?

By Albertfermat (Albertfermat) on Friday, July 09, 2004 - 07:04 pm: Edit

they r pretty easy, especially if u have done all of them

By Feuler (Feuler) on Friday, July 09, 2004 - 07:10 pm: Edit

In case other people want to try these but get stuck, here are a couple hints. These particular problems aren't too difficult, as AIME problems go.

1- Factor a^2-b^2.

2- Use prime factorization.

3- Think about what values the comman difference can be, keeping in mind that no two members of the sequence can share any prime factors.