Fascinating Facts about the number 2010
2010 = 1+2-(3-4-5)*6*7*8-9
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.