### 805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Today’s puzzle might get a little tricky just before it is completed:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

- 805 is a composite number.
- Prime factorization: 805 = 5 x 7 x 23
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 805 has exactly 8 factors.
- Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
- Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
- 795 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

- 483-644-805, which is 3-4-5 times 161

805 can be written as the sum of three squares four ways:

- 25² + 12² + 6² = 805
- 24² + 15² + 2² = 805
- 20² + 18² + 9² = 805
- 18² + 16² + 15² = 805

## Recent Comments