# Fun Facts About the Number 210

The number two hundred ten might not seem so remarkable at first glance, but this integer is surprisingly rich in mathematical properties. It also has much significance in pop culture, art, history, and religion. For example, in the Roman Empire, the year 210 AD was known as the year of the Consulship of Faustus and Rufinus. 210 BC was the year the Terracotta Warriors were made for the tomb of Qin Shi Huang, the first emperor of China who died that year, and who famously searched for an immortality elixir in the years leading up to his demise. In the United States, 210 is the area code for the region around San Antonio, Texas. Here are more fun facts about the number 210.

## Triangular Number Facts about 210

Triangular numbers are the sum of consecutive positive integers starting with 1. The Nth triangular number is the sum 1 + 2 + 3 + ... + N. Here are some triangular number facts involving 210.

- 210 is the 20th triangular number since 210 = 1 + 2 + 3 + ... + 19 + 20.
- 210 is a rare example of a triangular number that is also twice a triangular number. I.e., 210 = 2*105, and 105 is also a triangular number. Other triangular numbers that are twice a triangular number include 6 = 2*3 and 7140 = 2*3570.
- 210 is a rare example of a number that is both pentagonal and triangular. Although there are infinitely many triangular-pentagonal numbers, they are very spread out. The first several triangular-pentagonal numbers are 1, 210, 40755, 7906276, 153377680,...
- 210 is the product of two triangular numbers greater than 1, i.e., 10*21 = 210. There are infinitely many triangular numbers that are the product of two smaller triangular numbers. Some other examples are 36 = 6*6, 45 = 3*15, and 526851 = 171*3081.
- 210 can be written as the sum of two triangular numbers: 105 + 105 = 210. It can also be written as the sum of three triangular numbers in several different ways, for example, 210 = 190 + 10 + 10 and 210 = 171 + 36 + 3. Almost all triangular numbers can be expressed as the sum of three triangular numbers.

## 210 in Culture and Science

- July 29 is the 210th day of the year in non-leap years, and July 28 is the 210th day of the year in leap years.
- The 210th episode of The Simpsons was "Lisa Gets an A" in which Lisa cheats on a test and gets an A+++.
- The 210th episode of South Park was "HumancentiPad."
- The 210th episode of All in the Family was the last episode of the series.
- The 210th episode of Beverly Hills 90
was "Aloha Beverly Hills: Part 2." One of the show's stars, Luke Perry, received 210 fan letters per week.**210** - Washing your car by hand burns about 210 calories.
- Rearranging the furniture in your living room also burns about 210 calories.
- The boiling point of ammonium nitrate is 210 degrees Celsius.
- 210 is half of 420.

## Number Theoretic Facts about 210

- 210 is the product of the first four primes 2, 3, 5, and 7. The product of the first N primes is called the Nth primorial. The first 10 primorial numbers are 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, and 6469693230.
- 210 is a pronic number, meaning it is the product of two consecutive numbers. In this case, 210 = 14*15. The first several pronic numbers are 2, 6, 12, 20, 30, 42, 56, 72,...
- 210 is also the product of three consecutive integers; in this case, 210 = 5*6*7. The only other positive integer that is both pronic and the product of three consecutive integers is 6.
- 210 is the 12th pentagonal number.
- The sum of the squares of the factors of 12 is 210. Observe, 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2 = 210.
- 210 is a Harshad-Niven number, meaning it is divisible by the sum of its digits. 210/(2+1+0) = 70.
- 210 is also divisible by the sum of the squares of its digits: 210/(2^2 + 1^2 + 0^2) = 42.
- 210 is a number whose digits are in sequential order. There are only 90 such numbers.
- 210 is the number of ways to choose 4 distinct objects out of a set of 10 distinct objects.
- 210 is also the number of ways to choose 2 distinct objects out of a set of 21 distinct objects.
- There are 210 to partition a set of 22 elements into subsets that all have more than one element.
- The sum of all the factors of 210 is 576, which equals 24 squared.
- The sum of all the factors of 104 equals 210, and the sum of all the factors of 116 also equals 210. These are the only two numbers whose sum of divisors equals 210.
- 210 is an idoneal number, which means it cannot be written in the form a*b + b*c + c*a for any integers a, b, c. There are only 65 known idoneal numbers. In 1973 it was proven that there are at most 66 idoneal numbers, meaning that either the list of 65 is complete or it is missing one other number.

## Comments

Hi C G, 210 is a book number, assigned to me, that I had to write beside my signature in my 22 years on the fire department. When I first saw it in your title, excitement beamed over me. I see that my number had more significance to it. You're sharp! Thank you much for sharing. Voted up!