My Favorite Two-Digit Number

 

Most people do not think to have a preference for one two-digit number above all others. Possibly they choose a number because it reminds them of a birthday, or an anniversary. Numerologists, however, try to find meaning in numbers, where such meaning is probably questionable. Douglas Adams considered "42" to be the ultimate Answer to Life, the Universe, and Everything. I simply look for interesting arithmetic and other "uses" of a number. My favorite is "72".

 

1. 72 is the product of 2^3 (2 to the third power) and 3^2 (3 to the 2nd power). The numbers 2 and 3 are the first two prime numbers. Reversing the (base 10) digits of 72 gives 27. 27 is 3^3 (3 to the 3rd power).

2. 72 is the product of 8 and 9. The numbers 8 and 9 are the largest single-digit numbers (in base 10)... one is a perfect cube (2*2*2) and one is a perfect square (3*3).

3. 72 has 10 positive integer divisors, not counting itself and 1: 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36. No two-digit number (in base 10) has more divisors.

4. 72 is the sum of a subset of its proper divisors (12+24+36), and thus is a "semiperfect" number.

5. The digits "7" and "2" are, respectively, the largest and smallest single-digit prime numbers (in base 10).

6. 71 and 73, the integers immediately above and below 72, are both prime numbers. 72 is the largest two-digit number (in base 10) for which this is true.

7. 72 is divisible by six different single-digit (base 10) integers, which makes it a very convenient number of things to have, if the things must be shared equally among a small group of people.

8. 72 is twice as large as 36, which is a perfect square (6*6). 72 is half as large as 144, which is also a perfect square (12*12). 72 is also one third as large as 216, which is a perfect cube (6*6*6).

9. 72 is the sum of 4 consecutive primes (13+17+19+23), as well as the sum of six consecutive primes (5+7+11+13+17+19). It is the sum of two primes in six different ways: 5+67, 11+61, 13+59, 19+53, 29+43, and 31+41. It is the sum of three primes in five different ways: 2+3+67, 2+11+59, 2+17+53, 2+23+47, and 2+29+41. It is the sum of four primes in 57 different ways, and the sum of four distinct primes in 25 different ways.

10. The digits "7" and "2" are, respectively, odd and even. Something for everyone.

11. 72 mod 7 is 2. Of course, 44 other two-digit (base 10) numbers could say the same... 43 mod 4 is 3, etc.

12. 7 minus 2 is 5, and 5 is a prime number between 2 and 7. While other two-digit (base 10) numbers, like 41, 83, and 92, have "prime digit separation", these other two-digit numbers are not composed only of prime digits.

13. 72 to the 5th power is equal to the sum of five smaller 5th powers (19, 43, 46, 47, and 67). I have no idea how this was discovered to be true. 72 to the 3rd power is also equal to the sum of 3 smaller 3rd powers in 3 different ways: (8, 48, 64), (34, 39, 65), and (36, 48, 60).

14. There are 360 degrees in a circle, which is 5 times 72. Six dozen (half of a gross) is 72. Three days contain 72 hours. One fathom (which is 6 feet) contains 72 inches.

15. On a standard typewriter keyboard, the "7" key is above and to the left of the "U" key. The "2" key is above and to the left of the "W" (double-U) key.

16. 72 is evenly divisible by the sum of its (base 10) digits: 72 / (7+2) = 8. Only 23 of the 90 two-digit numbers have this property.

17. I became a teenager in 1972, and I hope I live to see 2072. 1972 was a leap year and was the longest year ever, as two leap seconds were added during it.

18. 72 appeared on the torsos of Carlton Fisk, Dan Dierdorf, and Ed “Too Tall” Jones, if you care about sports.