When it comes to applying our tried and true trick for divisibility by nine to the number 801, zero is just a place holder. Thus, since 81 is divisible by 9, so is 801. Adding up its digits was hardly necessary.

• 801 is a composite number.
• Prime factorization: 801 = 3 x 3 x 89, which can be written 801 = (3^2) x 89
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 801 has exactly 6 factors.
• Factors of 801: 1, 3, 9, 89, 267, 801
• Factor pairs: 801 = 1 x 801, 3 x 267, or 9 x 89
• Taking the factor pair with the largest square number factor, we get √801 = (√9)(√89) = 3√89 ≈ 28.301943396.

Would you be surprised to know the following division facts?

• 81 ÷ 3 = 27
• 801 ÷ 3 = 267
• 8001 ÷ 3 = 2667
• 80001 ÷ 3 = 26667 and so forth. The number of 6’s in the quotient is the same as the number of 0’s in the dividend!

Here are some more predictable division facts:

• 81 ÷ 9 = 9
• 801 ÷ 9 = 89
• 8001 ÷ 9 = 889
• 80001 ÷ 9 = 8889 and so forth. You guessed it! The number of 8’s in the quotient is the same as the number of 0’s in the dividend!

Even though you can’t see 81 in this puzzle with all perfect square clues, it isn’t difficult to see where 9 × 9 and 81 belong:

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

801 is a palindrome in three bases:

• 1441 BASE 8 because 1(8^3) + 4(8^2) + 4(8) + 1(1) = 801
• 2D2 BASE 17 D is 13 base 10 because 2(289) + 13(17) = 2(1) = 801
• 171 BASE 25 because 1(25²) + 7(25) + 1(1) =801

801 is the sum of two squares:

• 24² + 15² =801

So it follows that 801 is the hypotenuse of a Pythagorean triple:

• 351-720-801 which is 9 times 39-80-89

801 is the sum of three squares TEN ways:

1. 28² + 4² + 1² = 801
2. 27² + 6² + 6² = 801
3. 26² + 11² + 2² =801
4. 26² + 10² + 5² = 801
5. 24² + 12² + 9² = 801
6. 23² + 16² + 4² = 801
7. 22² + 14² + 11² = 801
8. 21² + 18² + 6² = 801
9. 20² + 20² + 1² = 801
10. 17² + 16² + 16² = 801

OEIS.org gives us this last fun fact:

801 = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10).