8 – 0 + 3 = 11, so 803 is divisible by 11.

• 803 is a composite number.
• Prime factorization: 803 = 11 x 73
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 803 has exactly 4 factors.
• Factors of 803: 1, 11, 73, 803
• Factor pairs: 803 = 1 x 803 or 11 x 73
• 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

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

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

• 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

• 27² + 7² + 5² = 803
• 25² + 13² + 3² = 803
• 23² + 15² + 7² = 803
• 21² + 19² + 1² = 803
• 19² + 19² + 9² = 803
• 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

• 263 + 269 + 271 = 803, that’s 3 consecutive primes.
• 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
• 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

• 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
• 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

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).

Since 88, its last two digits, are divisible by 4, we know that 788 and every other whole number ending in 88 is divisible by 4.

I learned the following fascinating fact about these six numbers starting with 788 from OEIS.org:

788 is also palindrome 404 in BASE 14. Note that 4(196) + 0(14) + 4(1) = 788.

788 is the hypotenuse of Pythagorean triple 112-780-788 which is 28-195-197 times 4.

Print the puzzles or type the solution on this excel file: 10-factors-788-794

• 788 is a composite number.
• Prime factorization: 788 = 2 x 2 x 197, which can be written 788 = (2^2) x 197
• 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 788 has exactly 6 factors.
• Factors of 788: 1, 2, 4, 197, 394, 788
• Factor pairs: 788 = 1 x 788, 2 x 394, or 4 x 197
• Taking the factor pair with the largest square number factor, we get √788 = (√4)(√197) = 2√197 ≈ 28.071338.

• 710 is a composite number.
• Prime factorization: 710 = 2 x 5 x 71
• 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 710 has exactly 8 factors.
• Factors of 710: 1, 2, 5, 10, 71, 142, 355, 710
• Factor pairs: 710 = 1 x 710, 2 x 355, 5 x 142, or 10 x 71
• 710 has no square factors that allow its square root to be simplified. √710 ≈ 26.645825.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are few more facts about 710:

710 can be written as the sum of consecutive numbers several ways:

• 176 + 177 + 178 + 179 = 710; that’s four consecutive numbers.
• 140 + 141 + 142 + 143 + 144 = 710; that’s five consecutive numbers.
• 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 710; that’s 20 consecutive numbers.

710 is the sum of the twenty prime numbers from 3 to 73.

Because 5 is one of its factors, 710 is the hypotenuse of Pythagorean triple 426-568-710. What is the greatest common factor of those three numbers?

710 is palindrome 868 in BASE 9; note that 8(81) + 6(9) + 8(1) = 710.

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• 705 is a composite number.
• Prime factorization: 705 = 3 x 5 x 47
• 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 705 has exactly 8 factors.
• Factors of 705: 1, 3, 5, 15, 47, 141, 235, 705
• Factor pairs: 705 = 1 x 705, 3 x 235, 5 x 141, or 15 x 47
• 705 has no square factors that allow its square root to be simplified. √705 ≈ 26.551836.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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What else can I tell you about the number 705?

Because 5 is one of its factors, 705 is the hypotenuse of the Pythagorean triple 423-564-705. What is the greatest common factor of those three numbers?

705 is the sum of consecutive numbers several different ways:

• 352 + 353 = 705; (2 consecutive numbers)
• 234 + 235 + 236 = 705; (3 consecutive numbers)
• 139 + 140 + 141 + 142 + 143 = 705; (5 consecutive numbers)
• 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 = 705; (15 consecutive numbers)

705 is palindrome 1A1 in base 22; note that A is equivalent to 10 base 10, 22² = 484, and 1(484) + 10(22) + 1(1) = 705.

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• 704 is a composite number.
• Prime factorization: 704 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 11, which can be written 704 = (2^6) ⋅ 11
• The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 ⋅ 2 = 14. Therefore 704 has exactly 14 factors.
• Factors of 704: 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704
• Factor pairs: 704 = 1 x 704, 2 x 352, 4 x 176, 8 x 88, 11 x 64, 16 x 44, or 22 x 32
• Taking the factor pair with the largest square number factor, we get √704 = (√64)(√11) = 8√11 ≈ 26.532998.

This Level 1 puzzle is not difficult at all:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more thoughts about the number 704:

7 – 0 + 4 = 11 so 704 is divisible by 11.

Here’s a fun multiplication fact: 8 x 88 = 704

Also 704 is a palindrome in two different bases:

• 1I1 in BASE 19; note 19² = 361, I is equivalent to 18 base 10, and 1(361) + 18(19) + 1(1) = 704
• MM in BASE 31; note M is equivalent to 22 base 10, and 22(31) = 22(1) = 704

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687 is made from three consecutive numbers so it is divisible by 3.

687 is the hypotenuse of Pythagorean triple 180-663-687. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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• 687 is a composite number.
• Prime factorization: 687 = 3 x 229
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 687 has exactly 4 factors.
• Factors of 687: 1, 3, 229, 687
• Factor pairs: 687 = 1 x 687 or 3 x 229
• 687 has no square factors that allow its square root to be simplified. √687 ≈ 26.21068.

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Ricardo tweeted the solution for this one:

@findthefactors in late.. But resolved pic.twitter.com/1WOzkeUtPr

— riccardo (@ricca9380) November 25, 2015

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