# Fascinating Facts about the number 2010

A number of blog postings have arisen over the past week discussing fascinating facts about the number 2010. Check out the posts at MathNotations and 360. I read about this gem:

2010 = 1+2-(3-4-5)*6*7*8-9

on a Mathletics Facebook post. And Theasmet has taken that to a whole new level!

edited: See also MAA Number A Day

With my middle schoolers, I figured it would be too challenging to ask them to find an expression such as the one above, so I used it as an order of operations review. I provided 4 variants on the expression above (each changing a single symbol) and asked them to compute the answers, carefully, using order of operations. I gave them the hint that one of the problems had 2010 as a solution. It was a fun way to use that expression, and showed us the parts of order of operations that still need a little shoring up.

For homework, I asked them to come up with some fascinating facts about 2010. I asked some leading questions to get them rolling, namely:

Things to think about:

- How many factors does 2010 have?
- How can 2010 be expressed as a sum of squares? Of cubes? Of primes? Of other interesting kinds of numbers?
- Can 2010 be expressed as a sum of consecutive integers? Of consecutive odd or even integers?

Here are some of the things they came up with. (More to come…)

- 2010 has 16 factors
- The sum of its digits is 3 (and prime)
- it has 4 distinct prime factors
- The product of its digits is 0
- 2010 = 40^2 + 20^2 + 3^2 + 1^2
- 2010 = 44^2 + 7^2 + 5^2
- 2010 = 13^3 + -4^3 + -4^3 + -4^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
- 2010 = 127 + 128 + … + 141
- 2010 = 669 + 670 + 671
- 2010 = 668 + 670 + 672
- 2010 cannot be expressed as the sum of any two consecutive integers
- The closest two integers that have a sum of 2010 are 1004 and 1006
- The closest two primes that have a sum of 2010 are 1013 and 997
- The largest number of consecutive integers that have a sum of 2010 is 15 (not true — need to get her to think about the cases with negative integers included)
- The only square factor of 2010 is 1
- You can sum to 2010 using addition and only (multiple copies of) the first two cube numbers.

Hi, Mathmom, and welcome back!

I like what you did with the order of operations exercise, and your students came up with some great facts. I would like to try a similar exercise with Math Club, but my students are too young this year. I think it will be as much as they can handle just to find the prime factors.

Hi Denise,

I actually gave them the prime factors, though finding them is something that they know how to do. And we reviewed how you can calculate how many factors a number has, if you know the prime factorization.

How old are your Math Club kids? I did “fascinating facts” about 97 and 153 a couple of years ago with a slightly younger group. I think if you prime the pump with age/ability-appropriate suggestions, almost any age can do this activity. I’ve done variations on this with kids as young as 8 or 9yo.

My kids range from 3rd to 5th grade, and most of them are absolute newbies. They’ve only just learned what prime numbers are, so I was planning to spend part of this month’s lesson on defining and finding prime factors. They also tend to panic with big numbers, so 2010 will be a nice challenge for them: big enough to look scary, yet easy to factor.

Denise, you might try something like this with your Math Club kids — I used it with a group about at that level, and it worked out really well. It turned out to be one of my favorite things I’ve done with that age group.

“The largest number of consecutive integers that have a sum of 2010 is 15 (not true — need to get her to think about the cases with negative integers included)”

I actually am going to compete in the chapter mathcounts competition this morning, and doing these exercises was a good warmup- the longest string is -2009 to 2010, which when summed will give 2010. That is, 4020 numbers.

Eliot, good job! I hope Mathcounts goes/went well for you and you’ll be prepping for State next. :)

1+2-(3-4-5)*6*7*8-9

= -1017

2010 = 1+2-(3-4-5)*6*7*8-9 ==> That managed to get my attention!

So simple and elegant … and it leaves you with a taste of … why didn’t I think of that!

Q: How did Ted manage to get -1017?

BTW … just came across your page of Math Goodness.

Enjoy!