# 767 is the hypotenuse of a Pythagorean triple, but…

• 767 is a composite number.
• Prime factorization: 767 = 13 x 59
• 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 767 has exactly 4 factors.
• Factors of 767: 1, 13, 59, 767
• Factor pairs: 767 = 1 x 767 or 13 x 59
• 767 has no square factors that allow its square root to be simplified. √767 ≈ 27.69476.

Because 13 is one of its factors, 767 is the hypotenuse of Pythagorean triple 295-708-767, but 767 will never be part of the solution to one of these Pythagorean triple puzzles because it is greater than 100. Every side and hypotenuse in these puzzles must be a whole number less than 100. Together the numbers in the puzzle must form Pythagorean triples. Can you solve the puzzle?

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here’s a little more about the number 767:

767 is a palindrome in two different bases:

• 767 in BASE 10; note that 7(100) + 6(10) + 7(7) = 767.
• 2B2 in BASE 17 (B = 11 base 10); note that 2(289) + 11(17) + 2(1) = 767.

Wikipedia informs us that 767 is also the 9th Thabit number.

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# 766 and Level 6

• 766 is a composite number.
• Prime factorization: 766 = 2 x 383
• 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 766 has exactly 4 factors.
• Factors of 766: 1, 2, 383, 766
• Factor pairs: 766 = 1 x 766 or 2 x 383
• 766 has no square factors that allow its square root to be simplified. √766 ≈ 27.6767.

Here’s a level 6 puzzle for you to try:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here are a few more reasons to be interested in the number 766:

766 is the sum of the twelve prime numbers from 41 to 89.

766 is palindrome 23332 in BASE 4; note that 2(256) + 3(64) + 3(16) + 3(4) + 2(1) = 766.

766 is the sum of three squares six different ways:

• 27² + 6² + 1² = 766
• 26² + 9² + 3² = 766
• 21² + 18² + 1² = 766
• 21² + 17² + 6² = 766
• 21² + 15² + 10² = 766
• 19² + 18² + 9² = 766

766 is also the 18th centered pentagonal number because (5⋅17² + 5⋅17 + 2)/2 = 766.

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This table shows a way to solve puzzle #766 using logic:

# Reducible Square Roots up to √765

• 765 is a composite number.
• Prime factorization: 765 = 3 x 3 x 5 x 17, which can be written 765 = (3^2) x 5 x 17
• 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 765 has exactly 12 factors.
• Factors of 765: 1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765
• Factor pairs: 765 = 1 x 765, 3 x 255, 5 x 153, 9 x 85, 15 x 51, or 17 x 45
• Taking the factor pair with the largest square number factor, we get √765 = (√9)(√85) = 3√85 ≈ 27.658633.

765 is the 300th number whose square root can be reduced! Here are three tables with 100 reducible square roots each showing all the reducible square roots up to √765. When three or more consecutive numbers have reducible square roots, I highlighted them.

That’s 300 reducible square roots found for the first 765 counting numbers. 300 ÷ 765 ≈ 0.392, so 39.2% of the numbers so far have reducible square roots.

Today’s puzzle is a whole lot less complicated than all that, so give it a try!

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

Logical steps to find the solution are in a table at the bottom of the post.

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Here are some other fun facts about the number 765:

765 is made from three consecutive numbers so it is divisible by 3. The middle of those numbers is 6 so 765 is also divisible by 9.

765 can be written as the sum of two squares two different ways:

• 27² + 6² = 765
• 21² + 18² = 765

Its other two prime factors, 5 and 17, have a remainder of 1 when divided by 4 so 765² can be written as the sum of two squares FOUR different ways, two of which contain other numbers that use the same digits as 765. Also notice that 9 is a factor of each number in the corresponding Pythagorean triples.

• 117² + 756² = 765²
• 324² + 693² = 765²
• 360² + 675² = 765²
• 459² + 612² = 765²

765 can also be written as the sum of three squares four different ways:

• 26² + 8² + 5² = 765
• 22² + 16² + 5² = 765
• 20² + 19² + 2² = 765
• 20² + 14² + 13² = 765

765 is a palindrome in two different bases:

• 1011111101 BASE 2; note that 1(512) + 0(256) + 1(128) + 1(64) + 1(32) + 1(16) + 1(8) + 1(4) + 0(2) + 1(1) = 765.
• 636 BASE 11; note that 6(121) + 3(11) + 6(1) = 765.

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# 764 and Level 4

• 764 is a composite number.
• Prime factorization: 764 = 2 x 2 x 191, which can be written 764 = (2^2) x 191
• 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 764 has exactly 6 factors.
• Factors of 764: 1, 2, 4, 191, 382, 764
• Factor pairs: 764 = 1 x 764, 2 x 382, or 4 x 191
• Taking the factor pair with the largest square number factor, we get √764 = (√4)(√191) = 2√191 ≈ 27.6405499.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Every number that ends in 4 is divisible by 2.

Every number that ends in 64 is divisible by 4.

Every number that ends in 764 is NOT divisible by 8, but every number ending in 764 can be written as the sum of 8 consecutive numbers.

For example, 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 = 764.

The first four digits of √764  are 27.64.

Wikipedia tells us that 764 is one of only two 3-digit “telephone numbers“. (911 is NOT the other one.)

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# 763 and Level 3

It’s obvious that 763 is divisible by 7 so it is a composite number.

• 763 is a composite number.
• Prime factorization: 763 = 7 x 109
• 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 763 has exactly 4 factors.
• Factors of 763: 1, 7, 109, 763
• Factor pairs: 763 = 1 x 763 or 7 x 109
• 763 has no square factors that allow its square root to be simplified. √763 ≈ 27.6224546.

Now try solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s a little more about the number 763:

763 is the sum of consecutive numbers three different ways:

• 381 + 382 = 763; that’s 2 consecutive numbers.
• 106 + 107 + 108 + 109 + 110 + 111 + 112 = 763; that’s 7 consecutive numbers.
• 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 763; that’s 14 consecutive numbers.

763 is also the sum of consecutive prime numbers two different ways:

• 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 = 763; that’s 9 consecutive primes.
• 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 763; that’s 17 consecutive primes.

763 is the hypotenuse of a Pythagorean triple, and 420² + 637² = 763².

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

• 27² + 5² + 3² = 763
• 23² + 15² + 3² = 763

763^4 = 338,920,744,561, a number in which every digit appears at least one time. Stetson.edu informs us 763 is the smallest number whose 4th power can make that claim.

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# 762 and Level 2

• 762 is a composite number.
• Prime factorization: 762 = 2 x 3 x 127
• 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 762 has exactly 8 factors.
• Factors of 762: 1, 2, 3, 6, 127, 254, 381, 762
• Factor pairs: 762 = 1 x 762, 2 x 381, 3 x 254, or 6 x 127
• 762 has no square factors that allow its square root to be simplified. √762 ≈ 27.604347.

This level 2 puzzle isn’t very difficult:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s a little more about the number 762:

People who memorize the digits of pi have to stop someplace. Wikipedia informs us that one place to stop is known as the Feynman point which is 999999 beginning at pi’s 762nd decimal place. It is named after Richard Feynman who reportedly said in a lecture that he would like to recite from memory the digits of pi up to that point because he could then end the recitation by saying 999999 and so on. Perhaps he would even be able to make pi sound like a rational number? Be sure to check out the highlighted digits of both pi and tau that appear in a graphic in that article. There is also an explanation of how truly unusual a sequence of six repeating digits can be.

762 can be written as the sum of consecutive numbers three ways:

• 253 + 254 + 255 = 762; that’s 3 consecutive numbers. (254 × 3 = 762)
• 189 + 190 + 191 + 192 = 762; that’s 4 consecutive numbers.
• 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69  = 762; that’s 12 consecutive numbers.

762 can also be written as the sum of two consecutive prime numbers: 379 + 383 = 762, and as the sum of four consecutive prime numbers: 181 + 191 + 193 + 197 = 762.

762 is the sum of three squares three different ways:

• 25² + 11² + 4² = 762
• 23² + 13² + 8² = 762
• 20² + 19² + 1² = 762

762 is palindrome and repdigit 222 in BASE 19 because 2(19²) + 2(19) + 2(1) = 762.

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# 761 and Level 1

• 761 is a prime number.
• Prime factorization: 761 is prime.
• The exponent of prime number 761 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 761 has exactly 2 factors.
• Factors of 761: 1, 761
• Factor pairs: 761 = 1 x 761
• 761 has no square factors that allow its square root to be simplified. √761 ≈ 27.586228.

How do we know that 761 is a prime number? If 761 were not a prime number, then it would be divisible by at least one prime number less than or equal to √761 ≈ 27.6. Since 761 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 761 is a prime number.

There will be a lot about squares in today’s post, and all of the clues in today’s puzzle are perfect squares, too:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 761:

Because 761 is a prime number, there is only one way to write it as the sum of consecutive numbers: 380 + 381 = 761.

However, because it is a centered square number, 761 can be written as the sum of consecutive square numbers: 20² + 19² = 761.

And because it is the sum of two consecutive square numbers, it is also the hypotenuse of a primitive Pythagorean triple: 39-760-761 which was calculated using 20² – 19², 2(20)(19), 20² + 19².

Notice also that 20 + 19 = 39, ⌊39²/2⌋ = 760, and ⌈39²/2⌉ = 761. (Those peculiar parenthesis mean⌊round down to the closest whole number⌋ and ⌈round up to the closest whole number⌉.)

In the 39 x 39 graphic shown above there are 760 white squares and 761 squares with some color. Thus, looking at that graphic is one way to look at the primitive Pythagorean triple 39-760-761.

All of that gives us another way to know that 761 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 20² + 19² = 761 with 20 and 19 having no common prime factors, 761 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √761 ≈ 27.6. Since 761 is not divisible by 5, 13, or 17, we know that 761 is a prime number.

761 can also be expressed as the sum of three squares TEN ways:

• 27² + 4² + 4² = 761
• 26² + 9² + 2² = 761
• 26² + 7² + 6² = 761
• 25² + 10² + 6² = 761
• 24² + 13² + 4² = 761
• 24² + 11² + 8² = 761
• 23² + 14² + 6² = 761
• 22² + 14² + 9² = 761
• 21² + 16² + 8² = 761
• 19² + 16² + 12² = 761

761 is a palindrome in two different bases:

• 535 BASE 12; note that 5(144) + 3(12) + 5(1) = 761.
• 1I1 BASE 20 (I = 18 base 10); note that 1(400) + 18(20) + 1(1) = 761.

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# 760 Not a Misprint: One side is 84, the other 48. What are the other sides?

• 760 is a composite number.
• Prime factorization: 760 = 2 x 2 x 2 x 5 x 19, which can be written 760 = (2^3) x 5 x 19
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 760 has exactly 16 factors.
• Factors of 760: 1, 2, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760
• Factor pairs: 760 = 1 x 760, 2 x 380, 4 x 190, 5 x 152, 8 x 95, 10 x 76, 19 x 40, or 20 x 38
• Taking the factor pair with the largest square number factor, we get √760 = (√4)(√190) = 2√190 ≈ 27.5680975.

There is no misprint in this puzzle. One side really is 84 while the other side really is 48. Can you find the other sides?

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 760:

760 is the sum of consecutive numbers two different ways. (Two of its factor pairs show up in those ways.):

• 150 + 151 + 152 + 153 + 154 = 760; that’s 5 consecutive numbers.
• 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 = 760; that’s 19 consecutive numbers.

760 is the hypotenuse of a Pythagorean triple so that 456² + 608² = 760²

760 is the sum of three squares: 20² + 18² + 6² = 760.

760 is palindrome 1A1 in Base 23 because 1(23²) + 10(23) + 1(1) = 760.

Wikipedia informs us that 760 is the 23rd centered triangular number because (3⋅22² + 3⋅22 + 2)/2 = 760.

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Here’s the same puzzle with out the lists of triples:

# 759 and Level 6

• 759 is a composite number.
• Prime factorization: 759 = 3 x 11 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 759 has exactly 8 factors.
• Factors of 759: 1, 3, 11, 23, 33, 69, 253, 759
• Factor pairs: 759 = 1 x 759, 3 x 253, 11 x 69, or 23 x 33
• 759 has no square factors that allow its square root to be simplified. √759 ≈ 27.5499546.

Here’s today’s puzzle. A logical way to find its solution is at the end of the post:

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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What else can I say about the number 759?

759 can be written as the sum of consecutive numbers seven different ways:

• 379 + 380 = 759; that’s 2 consecutive numbers.
• 252 + 253 + 254 = 759; that’s 3 consecutive numbers.
• 124 + 125 + 126 + 127 + 128 + 129 = 759; that’s 6 consecutive numbers.
• 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 = 759; that’s 11 consecutive numbers.
• 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45  = 759; that’s 22 consecutive numbers.
• 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44  = 759; that’s 23 consecutive numbers.
• 7 + 8 + 9 + 10 + 11 + 12 + 13 + 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 = 759; that’s 33 consecutive numbers.

759 is also the sum of five consecutive prime numbers: 139 + 149 + 151 + 157 + 163 = 759.

759 is a palindrome in two bases:

• 3C3 BASE 14 (C is 12 base 10)
• NN BASE 32 (N is 23 base 10)

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# 758 and Level 5

• 758 is a composite number.
• Prime factorization: 758 = 2 x 379
• 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 758 has exactly 4 factors.
• Factors of 758: 1, 2, 379, 758
• Factor pairs: 758 = 1 x 758 or 2 x 379
• 758 has no square factors that allow its square root to be simplified. √758 ≈ 27.5317998.

I started to feel like I was getting behind blogging so I decided over the weekend that it is okay NOT to blog everyday. I didn’t have time to post this puzzle until today, and I have a level 6 puzzle and another whole set of puzzles to post as well. I decided I don’t need to catch up. I’ll post them all soon enough. A logical way to find the solution is in a table at the end of the post.

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s more about the number 758:

758 is the sum of four consecutive numbers: 188 + 189 + 190 + 191 = 758.

758 is the sum of three squares six different ways:

• 27² + 5² + 2² = 758
• 26² + 9² + 1² = 756
• 23² + 15² + 2² = 758
• 22² + 15² + 7² = 758
• 21² + 14² + 11² = 758
• 19² + 19² + 6² = 758

758 is a palindrome in two different bases:

• 464 BASE 13; note that 4(13²) + 6(13) + 4(1) = 758.
• 262 BASE 18; note that 2(18²) + 6(18) + 2(1) = 758.

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