# The factors of the hundred numbers just before 1201

I’ve made a simple chart of the numbers from 1101 to 1200, but it’s packed with great information. It gives the prime factorization of each of those numbers and how many factors each of those numbers have. The numbers written with a pinkish hue are the ones whose square roots can be simplified. Notice that each of those numbers has an exponent in its prime factorization. I didn’t make a horserace from the amounts of factors this time because it isn’t a very close race. Nevertheless, you can guess which number appears most often in the “Amount of Factors columns” and see if your number would have won the race.

Now I’ll share some information about the next number, 1201. Notice the last entry in the chart above. It had so many factors that there weren’t very many left for 1201 to have. . .

• 1201 is a prime number.
• Prime factorization: 1201 is prime.
• The exponent of prime number 1201 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1201 has exactly 2 factors.
• Factors of 1201: 1, 1201
• Factor pairs: 1201 = 1 × 1201
• 1201 has no square factors that allow its square root to be simplified. √1201 ≈ 34.65545

How do we know that 1201 is a prime number? If 1201 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1201 ≈ 34.7. Since 1201 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1201 is a prime number. Even though it doesn’t have many factors, 1201 is still a fabulous number:

25² + 24² = 1201

1201 is the 25th Centered Square Number because 25² + 24² = 1201, and 24 and 25 are consecutive numbers: 1201 is the hypotenuse of a primitive Pythagorean triple:
49-1200-1201 calculated from 25² – 24², 2(25)(24), 25² + 24²

Here’s another way we know that 1201 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 24² = 1201 with 25 and 24 having no common prime factors, 1201 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1201 ≈ 34.7. Since 1201 is not divisible by 5, 13, 17, or 29, we know that 1201 is a prime number.

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