# 1061 and Level 5

Study the clues in the puzzle below. If you begin with the right set of clues, the puzzle can be solved quite easily, but if you don’t, you might get tripped up. Good luck! Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

Here are a few facts about the number 1061:

• 1061 is a prime number. 1061 and 1063 are twin primes.
• Prime factorization: 1061 is prime.
• The exponent of prime number 1061 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1061 has exactly 2 factors.
• Factors of 1061: 1, 1061
• Factor pairs: 1061 = 1 × 1061
• 1061 has no square factors that allow its square root to be simplified. √1061 ≈ 32.57299

How do we know that 1061 is a prime number? If 1061 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1061 ≈ 32.6. Since 1061 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1061 is a prime number. 1061 is the sum of the 17 prime numbers from 29 to 101, and it also is the sum of these three consecutive prime numbers: 349 + 353 + 359 = 1061

31² + 10² = 1061 so 1061 is the hypotenuse of a Pythagorean triple:
620-861-1061, a primitive calculated from 2(31)(10), 31² – 10², 31² + 10²

Here’s another way we know that 1061 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 10² = 1061 with 31 and 10 having no common prime factors, 1061 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1061 ≈ 32.6. Since 1061 is not divisible by 5, 13, 17, or 29, we know that 1061 is a prime number.

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