## A Multiplication Based Logic Puzzle

### 695 and Level 1

• 695 is a composite number.
• Prime factorization: 695 = 5 x 139
• 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 695 has exactly 4 factors.
• Factors of 695: 1, 5, 139, 695
• Factor pairs: 695 = 1 x 695 or 5 x 139
• 695 has no square factors that allow its square root to be simplified. √695 ≈ 26.36285.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are a few more facts about the number 695:

695 is the sum of the eighteen prime numbers from 7 to 73.

695 is also the hypotenuse of the Pythagorean triple 417-556-695. What is the greatest common factor of those three numbers?

137 + 138 + 139 + 140 + 141 = 695 so it is the sum of 5 consecutive numbers.

Since 695 is odd, it is the sum of two consecutive numbers, 347 and 348.

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### 694 and Level 6

• 694 is a composite number.
• Prime factorization: 694 = 2 x 347
• 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 694 has exactly 4 factors.
• Factors of 694: 1, 2, 347, 694
• Factor pairs: 694 = 1 x 694 or 2 x 347
• 694 has no square factors that allow its square root to be simplified. √694 ≈ 26.34388.

Level 6 puzzles aren’t easy, but do give it a try! There are hints at the bottom of the page.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here are some ways that 694 is a special number:

694 is the 22nd centered triangular number because it is the sum of these consecutive triangular numbers 210, 231, 253.

Notice that 21(22)/2 is 231, the number in the middle of those consecutive triangular numbers. Also notice that 3(231) + 1 = 694.

694 BASE 10 becomes palindrome 1C1 in BASE 21 because C Base 21 is equivalent to 12 Base 10, and 1(21²) + 12(21) + 1(1) = 694.

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### Divisibility Tricks Applied to 693

• 693 is a composite number.
• Prime factorization: 693 = 3 x 3 x 7 x 11, which can be written 693 = (3^2) x 7 x 11
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 693 has exactly 12 factors.
• Factors of 693: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693
• Factor pairs: 693 = 1 x 693, 3 x 231, 7 x 99, 9 x 77, 11 x 63, or 21 x 33
• Taking the factor pair with the largest square number factor, we get √693 = (√9)(√77) = 3√77 ≈ 26.324893

Some quick divisibility tricks applied to the number 693:

1. Every counting number is divisible by 1
2. 693 is not even so it isn’t divisible by 2
3. Every digit of 693 is divisible by 3, so 693 is divisible by 3
4. Since it isn’t divisible by 2, it isn’t divisible by 4
5. 693 doesn’t end in a 5 or 0, so it’s not divisible by 5
6. 693 is divisible by 3 but not by 2 so it isn’t divisible by 6
7. 69 – 2(3) = 63, a multiple of 7 so 693 is divisible by 7
8. Since it isn’t divisible by 2 or 4, it can’t be divisible by 8
9. 6 + 9 + 3 = 18, a multiple of 9 so 693 is divisible by 9
10. Since the last digit of 693 isn’t 0, it is not divisible by 10
11. 69 + 3 = 0, so 693 is divisible by 11

The divisibility tricks that worked on the number 693 are quite easy to see on the outside of this factor cake.

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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693 is a palindrome in several bases:

• 1010110101 BASE 2; note 1(512) + 0(256) + 1(128) + 0(64) + 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1) = 693
• 3113 BASE 6; note 3(216) + 1(36) + 1(6) + 3(1) = 693
• 414 BASE 13; note 4(169) + 1(13) + 4(1) = 693
• 313 BASE 15; note 3(225) + 1(15) + 3(1) = 693

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Ricardo tweeted his work for this puzzle, too.

### 692 Happy Thanksgiving!

• 692 is a composite number.
• Prime factorization: 692 = 2 x 2 x 173, which can be written 692 = (2^2) x 173
• 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 692 has exactly 6 factors.
• Factors of 692: 1, 2, 4, 173, 346, 692
• Factor pairs: 692 = 1 x 692, 2 x 346, or 4 x 173
• Taking the factor pair with the largest square number factor, we get √692 = (√4)(√173) = 2√173 ≈ 26.30589.

Today is Thanksgiving in the United States. Regardless of where we live, there are reasons to be grateful today and every other day. Corrie ten Boom was grateful even when her circumstances were horrible. Here’s a Thanksgiving themed puzzle to solve:

Sometimes color in a puzzle is a distraction. Here’s the same puzzle minus the color:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here’s a little more about the number 692:

Because 173 is one of its factors, 692 is the hypotenuse of Pythagorean triple 208-660-692. What is the greatest common factor of those three numbers?

692 is a palindrome in 2 bases:

• 221122 BASE 3; note that 2(243) + 2(81) + 1(27) + 1(9) + 2(3) + 2(1) = 692
• 848 BASE 9; note that 8(81) + 4(9) + 8(1) = 692

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### 691 and Level 3

• 691 is a prime number.
• Prime factorization: 691 is prime and cannot be factored.
• The exponent of prime number 691 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 691 has exactly 2 factors.
• Factors of 691: 1, 691
• Factor pairs: 691 = 1 x 691
• 691 has no square factors that allow its square root to be simplified. √691 ≈ 26.2868789.

How do we know that 691 is a prime number? If 691 were not a prime number, then it would be divisible by at least one prime number less than or equal to √691 ≈ 26.3. Since 691 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 691 is a prime number.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here’s a little more about 691:

691 is the sum of the thirteen prime numbers from 29 to 79.

691 is equivalent to palindrome 171 in BASE 23. Note that 1(23²) + 7(23) + 1(1) = 691.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

### When is 690 a Palindrome? In Base 16 and Base 29.

• 690 is a composite number.
• Prime factorization: 690 = 2 x 3 x 5 x 23
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 690 has exactly 16 factors.
• Factors of 690: 1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690
• Factor pairs: 690 = 1 x 690, 2 x 345, 3 x 230, 5 x 138, 6 x 115, 10 x 69, 15 x 46, or 23 x 30
• 690 has no square factors that allow its square root to be simplified. √690 ≈ 26.267851

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here is a little more about the number 690:

690 is the sum of the six prime numbers from 103 to 131. Do you know what all of those prime numbers are?

690 is also the hypotenuse of Pythagorean triple 414-552-690. What is the greatest common factor of those three numbers?

In BASE 10 we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For bases smaller than 10 we use only the digits less than that base. For bases greater than 10, we might use A to represent 10, B to represent 11, and so forth all the way to Z representing 35 in BASE 36.

690 is a palindrome in two bases that require us to use letters of the alphabet to represent it:

• 2B2 in BASE 16; note that 2(256) + 11(16) + 2(1) = 690
• NN in BASE 29; note that 23(29) + 23(1) = 690. (N is the 14th letter of the alphabet and 14 + 9 = 23)

NN looks like it is divisible by 11, but remember that 11 base 29 is the same as 30 in base 10.

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### What Makes 689 Amazing?

689 is an amazing number for several reasons. I decided to make graphics to illustrate many of those ways. (689’s factoring information is at the end of this post.)

689 is the sum of consecutive prime numbers 227, 229, and 233.

Also 689 is the sum of the primes from 83 to 109. Do you know what those 7 prime numbers are?

Stetson.edu informs us that 689 is the smallest number that can be expressed as the sum of three different square numbers NINE ways. I decided to figure out what those nine ways are and make this first graphic to share with you:

Note: 614 can also be expressed as the sum of 3 squares 9 different ways, but one of those ways is 17² + 17² + 6² = 614, and that duplicates 17² in the same sum.

689 is the same number when it is turned upside down. Numbers with that characteristic are called Strobogrammatic numbers.

689 BASE 10 isn’t a palindrome, but 373 BASE 14 is; note that 3(196) + 7(14) + 3(1) = 689

Both of 689’s prime factors have a remainder of 1 when divided by 4, so they are hypotenuses of Pythagorean triples. That fact also means 689 can be expressed as the sum of two square numbers TWO different ways, and it makes 689 the hypotenuse of FOUR Pythagorean triples.  Can you tell by looking at the graphic which two are primitive and which two aren’t?

689 is the sum of consecutive numbers three different ways:

• 344 + 345 = 689; that’s 2 consecutive numbers.
• 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 689; that’s 13 consecutive numbers.
• 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 689; that’s 26 consecutive numbers.

Now you have a few reasons why 689 is an amazing number. 13 and 53 were part of some of those reasons so it shouldn’t surprise anyone to see 13 and 53 pop up in its factoring information, too:

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• 689 is a composite number.
• Prime factorization: 689 = 13 x 53
• 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 689 has exactly 4 factors.
• Factors of 689: 1, 13, 53, 689
• Factor pairs: 689 = 1 x 689 or 13 x 53
• 689 has no square factors that allow its square root to be simplified. √689 ≈ 26.248809.