It’s exciting times around PWN HQ—lots of things going on. 2014 should be a fun year for SAT prep. That has nothing to do with this contest, of course. I just like to open these contest posts with a little friendly chatter. I bet nobody even reads this stuff. :/ ANYWAY, here’s a challenge question!

For all positive integers n, let ↭n equal the number of unique prime factors of n. For example, ↭12 = 2, because there are 2 unique prime factors of 12: 2 and 3. If a is a positive integer less than 500,000, what is the greatest possible value of ↭a?

Post your answer in the comments—the first correct answer from someone who hasn’t won a contest before wins a Math Guide! Comments are set to require moderation until tomorrow to add a little suspense into the contest.

All the usual contest rules apply. One recent addition to the contest rules I’d like to draw your attention to:

• No answer changing. When you post a comment, I get an email with that comment, and that’s what I use to judge the contest. Edits applied to your comment later don’t count, even if the edit occurs before someone else wins. Don’t post your comment until you’re sure of your answer.
Good luck!

UPDATE: Wow—I thought this would be tougher for you guys! Congrats to everyone who got it right, especially to Nick, who got his answer in first. Explanation follows below the cut…

The key to getting this question right is recognizing that the smaller the prime factors you use, the more you’ll be able to fit in before your product exceeds 500,000. Therefore, all you need to do to solve this problem is test out the incremental products of the smallest prime numbers!

2 × 3 = 6
2 × 3 × 5 = 30
2 × 3 × 5 × 7 = 210
2 × 3 × 5 × 7 × 11 = 2,310
2 × 3 × 5 × 7 × 11 × 13 = 30,030
2 × 3 × 5 × 7 × 11 × 13 × 17 = 510,510    ← Oops, too big!

What we’ve just shown there is the smallest number that can have 2 unique prime factors is 6, the smallest number that can have 3 unique prime factors is 30, …, the smallest number that can have 7 unique prime factors is 510,510. Therefore, no positive integer less than 500,000 can have more than 6 unique prime factors.

Joshua Marsh says:

I think the answer is 6. The maximum value will be when you multiply unique prime numbers together. If you multiply the 7 smallest primes together, the answer is bigger than 500,000.

troldtog says:

The answer is 6. The smallest number to have this many prime factors is 2*3*5*7*11*13=30030

Joshua Marsh says:

I commented a bit ago, but I’m not sure if it showed up because it says there are zero comments. I really would want that book, so I’d like to put here that I got the answer of 6 by combining all of the smallest prime numbers until I broke 500,000.

Guest says:

never mind i know i was wrong i was supposed to multiply the prime numbers until i got to the largest number less than 500,000 i know i can’t change the answer
but thank you