# 773 and Level 6

• 773 is a prime number.
• Prime factorization: 773 is prime.
• The exponent of prime number 773 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 773 has exactly 2 factors.
• Factors of 773: 1, 773
• Factor pairs: 773 = 1 x 773
• 773 has no square factors that allow its square root to be simplified. √773 ≈ 27.8028775. How do we know that 773 is a prime number? If 773 were not a prime number, then it would be divisible by at least one prime number less than or equal to √773 ≈ 27.8. Since 773 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 773 is a prime number.

Here is today’s puzzle for you to try to solve: Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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What else is special about the number 773?

22² + 17² = 773 so 773 is the hypotenuse of the primitive Pythagorean triple 195-748-773 which was calculated using 22² – 17², 2(17)(22), 22² + 17².

Thus 195² + 748² + 773².

773 is also the sum of three squares six different ways:

• 26² + 9² + 4² = 773
• 25² + 12² + 2² = 773
• 24² + 14² + 1² = 773
• 23² + 12² + 10² = 773
• 22² + 15² + 8² = 773
• 20² + 18² + 7² = 773

773 is a palindrome in two other bases:

• 545 BASE 12, note that 5(144) + 4(12) + 5(1) = 773
• 3D3 BASE 14 (D = 13 base 10); note that 3(196) + 13(14) + 3(1) = 773

Here’s another way we know that 773 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 22² + 17² = 773 with 22 and 17 having no common prime factors, 773 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √773 ≈ 27.8. Since 773 is not divisible by 5, 13, or 17, we know that 773 is a prime number.

————————————- ## 2 thoughts on “773 and Level 6”

1. davescarthin

I’ve done a lattice labyrinth for the sufficiently massive prime 773, based on number pair (17,22), 17^2 + 22^2 being equal to 773, as you point out, but don’t know how to send it. Best wishes, Dave Mitchell

• ivasallay

If you give me the link to the lattice labyrinth on your blog, I can cut and paste it onto this blog. Thanks.

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