1181 and Level 5

Some parts of this puzzle are easier than others, but it all still a lot of fun! Give it a try and enjoy yourself!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Now I’ll share some facts about the number 1181:

• 1181 is a prime number.
• Prime factorization: 1181 is prime.
• The exponent of prime number 1181 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1181 has exactly 2 factors.
• Factors of 1181: 1, 1181
• Factor pairs: 1181 = 1 × 1181
• 1181 has no square factors that allow its square root to be simplified. √1181 ≈ 34.36568

How do we know that 1181 is a prime number? If 1181 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1181 ≈ 34.4. Since 1181 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1181 is a prime number.

34² + 5² = 1181

1181 is the hypotenuse of a Pythagorean triple:
340-1131-1181 calculated from 2(34)( 5), 34² – 5², 34² + 5²

Here’s another way we know that 1181 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 5² = 1181 with 34 and 5 having no common prime factors, 1181 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1181 ≈ 34.4. Since 1181 is not divisible by 5, 13, 17, or 29, we know that 1181 is a prime number.

1181 is also palindrome 353 in BASE 19
because 3(19²) + 5(19) + 3(1) = 1181