### Today’s Puzzle:

Here’s one more Valentine’s puzzle for you to do. Enjoy!

### Factors of 1749:

1749 is divisible by 3 because 7 is half of 14 so any 3-digit number made from the digits 714 in any order is divisible by 3, and of course, 9 is divisible by 3.

1749 is divisible by 11 because 1 – 7 + 4 – 9 = -11, a multiple of 11.

- 1749 is a composite number.
- Prime factorization: 1749 = 3 × 11 × 53.
- 1749 has no exponents greater than 1 in its prime factorization, so √1749 cannot be simplified.
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1749 has exactly 8 factors.
- The factors of 1749 are outlined with their factor pair partners in the graphic below.

### More About the Number 1749:

1749 is the hypotenuse of a Pythagorean triple:

924-1485-1749 which is **33** times (28-45-**53**).

1749 is the difference of two squares in FOUR different ways:

875² – 874² = 1749,

293² – 290² = 1749,

85² – 74² = 1749, and

43² – 10² = 1749.

I really enjoy your information on numbers. Like Ramanujan you’re a friend of every number.

Thank you. He was much more of a genius than I could ever hope to be. I consider it an honor that you see some small way that I’m like him.

I am very interested in prime quadruplets. Is there any other way than checking every decade whose beginning number ends in 1, or multiplying primes whose factorization contains 5 or a power of 5?

That’s a very good question. I will have to think about it some more before I can respond.