## A Multiplication Based Logic Puzzle

### 815 and Level 1

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

• 489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

Print the puzzles or type the solution on this excel file: 12 factors 815-820

• 815 is a composite number.
• Prime factorization: 815 = 5 x 163
• 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 815 has exactly 4 factors.
• Factors of 815: 1, 5, 163, 815
• Factor pairs: 815 = 1 x 815 or 5 x 163
• 815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

### 807 and Level 1

What can I say about the number 807?

807 is palindrome 151 in BASE 26 because 1(26²) + 5(26) + 1(1) = 807.

Anything else? Well, I can figure out a few other things because 807’s has two prime factors, 3 and 269:

We can write ANY number (unless it’s a power of 2) as the sum of consecutive numbers in at least one way. 807 has three different ways to do that:

• 403 + 404 = 807 because 807 isn’t divisible by 2.
• 268 + 269 + 270 = 807 because it is divisible by 3.
• 132 + 133 + 134 + 135 + 136 + 137 = 807 since it is divisible by 3 but not by 6.

I know that one of 807’s factors, 269, is a hypotenuse of a Pythagorean triple, so 807 is also. Thus. . .

• (3·69)² + (3·260)² = (3·269)², or in other words, 207² + 780² = 807²

Since 807 has two odd sets of factor pairs, I know that 807 can be written as the difference of two squares two different ways:

• 136² – 133² = 807
• 404² – 403² = 807

I don’t usually do this, but today’s puzzle has something in common with 807. Can you tell what it is?

Print the puzzles or type the solution on this excel file: 10-factors 807-814

• 807 is a composite number.
• Prime factorization: 807 = 3 x 269
• 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 807 has exactly 4 factors.
• Factors of 807: 1, 3, 269, 807
• Factor pairs: 807 = 1 x 807 or 3 x 269
• 807 has no square factors that allow its square root to be simplified. √807 ≈ 28.4077454

### 801 and Level 1

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

Stetson.edu gives us this last fun fact:

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

### 795 a Level 1 Puzzle with ♥

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a when the keys are released. I added a few ‘s to today’s puzzle because Valentine’s day is almost here:

• 795 is a composite number.
• Prime factorization: 795 = 3 x 5 x 53
• 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 795 has exactly 8 factors.
• Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
• Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
• 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

Stetson.edu informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

• 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

• 69 – 792 – 795 which is 3 times (23 – 264 – 265)
• 288 – 741 – 795 which is 3 times (96 – 247 – 265)
• 420 – 675 – 795 which is 15 times (28 – 45 – 53)
• 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

• 25² + 13² + 1² = 795
• 25² + 11² + 7² = 795

### 788 and Level 1

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 Stetson.edu:

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.

### 782 and Level 1

• 782 is a composite number.
• Prime factorization: 782 = 2 x 17 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 782 has exactly 8 factors.
• Factors of 782: 1, 2, 17, 23, 34, 46, 391, 782
• Factor pairs: 782 = 1 x 782, 2 x 391, 17 x 46, or 23 x 34
• 782 has no square factors that allow its square root to be simplified. √782 ≈ 27.96426.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Stetson.edu informs us that the sum of the factors of 782 is not only a perfect square number but also a perfect 4th power.

Specifically, 1 + 2 + 17 + 23 + 34 + 46 + 391 + 782 = 1296, which happens to be 6^4.

23 is one of the factors of 782, and here are three interesting number facts about the number 782 that involve the number 23:

• 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 782. (That’s 23 consecutive numbers beginning with the number 23.)
• 782 is the 23rd pentagonal number because 23(3⋅23 – 1)/2 = 782.
• 782 is palindrome NN in BASE 33. (N is 23 base 10.) Note that 23(33) + 23(1) = 782

Because 17 is one of its factors, 782 is the hypotenuse of Pythagorean triple 368-690-782.

Thus, 368² + 690² = 782².

Also since 17 x 46 = 782, the greatest common factor of 368, 690 and 782 is 46.

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

• 27² + 7² + 2² = 782
• 26² + 9² + 5² = 782
• 25² + 11² + 6² = 782
• 22² + 17² + 3² = 782
• 19² + 15² + 14² = 782
• 18² + 17² + 13² = 782

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### 774 and Level 1

• 774 is a composite number.
• Prime factorization: 774 = 2 x 3 x 3 x 43, which can be written 774 = 2 x (3^2) x 43
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 774 has exactly 12 factors.
• Factors of 774: 1, 2, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774
• Factor pairs: 774 = 1 x 774, 2 x 387, 3 x 258, 6 x 129, 9 x 86, or 18 x 43
• Taking the factor pair with the largest square number factor, we get √774 = (√9)(√86) = 3√86 ≈ 27.820855.

Here is a factoring puzzle that is just right for beginners:

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s a little more about the number 774:

774 is the sum of consecutive prime numbers two different ways:

• 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 774; that’s 18 consecutive primes.
• 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 774; that’s 14 consecutive primes.

774 is the sum of three triangular numbers eight different ways:

• 666 + 105 + 3 = 774
• 630 + 78 + 66 = 774
• 561 + 210 + 3 = 774
• 528 + 231 + 15 = 774
• 528 + 210 + 36 = 774
• 465 + 231 + 78 = 774
• 378 + 351 + 45 = 774
• 378 + 276 + 120 = 774

774 is also the sum of three squares nine different ways:

• 27² + 6² + 3² = 774
• 26² + 7² + 7² = 774
• 25² + 10² + 7² = 774
• 23² + 14² + 7² = 774
• 22² + 17² + 1² = 774
• 22² + 13² + 11² = 774
• 21² + 18² + 3² = 774
• 18² + 15² + 15² = 774
• 17² + 17² + 14² = 774

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