# Unique Products on a Times Table

I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed numbers -- most of them are either primes >10 or multiples of those primes. I couldn't figure out a satisfying reason like that for 75, 84, 96 and 98. I feel like the number 42 should be calculable from the natural log of 100 and 10, or something, but I can't really figure it out.

Also, I counted the frequency of all the numbers represented. They all have either 1, 2, 3, or 4 instances. The six numbers that have only 1 instance are all squares -- those of 1, 5, 7, 8, 9, and 10. This makes sense -- squares don't fall into that 4x5 = 5x4 redundancy. The 23 numbers with 2 instances are kind of the default I guess. The four with 3 are the remaining squares -- 4, 9, 16, and 36. 4 and 9 make sense to me, as they are squares within the basic range of the times table, so they are hit by themselves, their square root, and 1. I'm having trouble accounting for why 16 and 36 have three representatives. I'm also having trouble seeing the pattern (or patterns) in the numbers with four representatives.

I ultimately want to find a way to, given a times table of any size, calculate whether a given number will appear, and how many times.

Also, I'm pretty sure that for times tables 7x7 and below, more than half of the possible numbers are represented on the table, but for times tables above 7x7, less than half are represented. Why is 7 the turning point?

• I don't know about exact calculations as you ask about. But, if probabilistic calculations interest you, you might check out the Erdos-Kac theorem (it's got a Wikipedia page) and related work of Erdos on probabilistic properties of prime factorizations. – Lee Mosher Jul 31 '14 at 17:55
• This is the multiplication table problem of Erdos. A preprint of a paper appearing in the Annals, the premier math journal, has Asymptotics on this. It appears at the bottom of pg 372 and the top of pg 373. – davidlowryduda Jul 31 '14 at 17:58
• – Gerry Myerson Aug 3 '14 at 0:27
• – Gerry Myerson Aug 3 '14 at 0:28

Consider the prime factorization of a number in the Times Table; better yet, let's look at any one of the mysterious missing numbers, say $84$
The prime factors of $84$ are:$$(2, 2, 3, 7)$$ So, the question is: Can you divide this set of factors into two sub-sets, such that the product of the factors in each subset is less than or equal to 10?
Clearly, the $7$ must stand alone; combining it with any other factor puts you outside the $1-10$ range. But that leaves $12$.
The same analysis explains the other unexplained missings: $75, 96, and 98$.