# 772 and Level 5

• 772 is a composite number.
• Prime factorization: 772 = 2 x 2 x 193, which can be written 772 = (2^2) x 193
• 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 772 has exactly 6 factors.
• Factors of 772: 1, 2, 4, 193, 386, 772
• Factor pairs: 772 = 1 x 772, 2 x 386, or 4 x 193
• Taking the factor pair with the largest square number factor, we get √772 = (√4)(√193) = 2√193 ≈ 27.78488798.

Here is a factoring puzzle to try:

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

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Here’s more about the number 772:

24² + 14² = 772 so 772 is the hypotenuse of a Pythagorean triple, and 380² + 672² = 772². 380-672-772 is calculated from 24² – 14², 2(24)(14) , 24² + 14².

22² + 12² + 12² = 772, making 772 the sum of three square numbers.

772 is also the sum of two consecutive prime numbers: 383 + 379 = 772.

Stetson.edu informs us that 772 is the smallest number that is the sum of three triangular numbers 21 different ways. I decided to find all those ways for myself and share them here. (If zero wasn’t named the zeroth triangular number, there would “only” be 20 ways.)

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# A regular 771-sided polygon can be constructed with a straightedge and compass.

Happy pi day to lovers of mathematics who happen to write their dates mm/dd/yyyy or mm/dd/yy (even for only one day a year)!

Lots will be written by others about pi today, but I’m going to write about the number 771 instead.

• 771 is a composite number.
• Prime factorization: 771 = 3 x 257
• 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 771 has exactly 4 factors.
• Factors of 771: 1, 3, 257, 771
• Factor pairs: 771 = 1 x 771 or 3 x 257
• 771 has no square factors that allow its square root to be simplified. √771 ≈ 27.76688675.

The prime factors of 771 are 3 and 257, both of which are Fermat primes.  Wikipedia informs us that a regular 771-sided polygon can therefore be constructed using only a straightedge and a compass. Today’s puzzle looks a little like a compass and a straightedge:

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

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Here is more about the number 771:

Like pi, √771 is irrational. The first few digits of √771 are 27.76688.

771 is the sum of three consecutive prime numbers: 251 + 257 + 263 = 771. One of those prime numbers is also a factor of 771.

Because 257 is one of its factors, 771 is the hypotenuse of a Pythagorean triple, and 96² + 765² = 771².

771 is also the sum of three squares four different ways. Notice that 11 and/or 19 appear in every one of those ways:

• 25² + 11² + 5² = 771
• 23² + 11² + 11² = 771
• 19² + 19² + 7² = 771
• 19² + 17² + 11² = 771

771 is a palindrome in FIVE different bases:

• 1100000011 BASE 2; note that 1(2^9) + 1(2^8) + 0(2^7) + 0(2^6) + 0(2^5) + 0(2^4) + 0(2^3) + 0(2^2) + 1(2^1) + 1(2^0) = 771.
• 30003 BASE 4; note that 3(4^4) + 0(4^3) + 0(4^2) + 0(4^1) + 3(4^0) = 771.
• 474 BASE 13; note that 4(169) + 7(13) + 4(1) = 771.
• 303 BASE 16; note that 3(256) + 0(16) + 3(1) = 771.
• 1D1 BASE 22 (D is 13 base 10); note that 1(22²) + 13(22) + 1(1) = 771.

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# 770 and Level 3

• 770 is a composite number.
• Prime factorization: 770 = 2 x 5 x 7 x 11
• 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 770 has exactly 16 factors.
• Factors of 770: 1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770
• Factor pairs: 770 = 1 x 770, 2 x 385, 5 x 154, 7 x 110, 10 x 77, 11 x 70, 14 x 55, or 22 x 35
• 770 has no square factors that allow its square root to be simplified. √770 ≈ 27.74887.

Here is a puzzle for you to solve:

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

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Because 5 is one of its factors, 770 is the hypotenuse of a Pythagorean triple, and 462² + 616² = 770².

770 is the sum of the squares of three consecutive numbers: 15² + 16² + 17² = 770.

770 can also be written as the sum of three squares seven other ways:

• 27² + 5² + 4² = 770
• 25² + 12² + 1² = 770
• 25² + 9² + 8² = 770
• 24² + 13² + 5² = 770
• 23² + 15² + 4² = 770
• 20² + 19² + 3² = 770
• 20² + 17² + 9² = 770

770 is palindrome MM in Base 34 (M = 22 base 10); note that 22(34) + 22(1) = 770.

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# 769 and Level 2

• 769 is a prime number.
• Prime factorization: 769 is prime.
• The exponent of prime number 769 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 769 has exactly 2 factors.
• Factors of 769: 1, 769
• Factor pairs: 769 = 1 x 769
• 769 has no square factors that allow its square root to be simplified. √769 ≈ 27.730849.

How do we know that 769 is a prime number? If 769 were not a prime number, then it would be divisible by at least one prime number less than or equal to √769 ≈ 27.7. Since 769 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 769 is a prime number.

Here is today’s puzzle:

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

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Here’s more about prime number 769:

25² + 12² = 769, and 769 is the hypotenuse of primitive Pythagorean triple 481-600-769 which was calculated from 25² – 12², 2(25)(12), 25² + 12².

Thus 481² + 600² = 769².

769 is also the sum of three squares five different ways.

• 27² + 6² + 2² = 769
• 24² + 12² + 7² = 769
• 21² + 18² + 2² = 769
• 20² + 15² + 12² = 769
• 18² + 18² + 11² = 769

769 is palindrome 181 in BASE 24; note that 1(24²) + 8(24) + 1(1) = 769.

Here’s another way we know that 769 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 25² + 12² = 769 with 25 and 12 having no common prime factors, 769 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √769 ≈ 27.7. Since 769 is not divisible by 5, 13, or 17, we know that 769 is a prime number.

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# 768 and Level 1

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

• 768 is a composite number.
• Prime factorization: 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3, which can be written 768 = (2^8) x 3
• The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 x 2 = 18. Therefore 768 has exactly 18 factors.
• Factors of 768: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768
• Factor pairs: 768 = 1 x 768, 2 x 384, 3 x 256, 4 x 192, 6 x 128, 8 x 96, 12 x 64, 16 x 48 or 24 x 32
• Taking the factor pair with the largest square number factor, we get √768 = (√256)(√3) = 16√3 ≈ 27.7128129.

Today’s puzzle is relatively easy:

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

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Here’s a few more facts about the number 768:

255 + 256 + 257 = 768. That is the ONLY way to write 768 as the sum of consecutive positive whole numbers.

There is also only one way to write 768 as the sum of three square numbers:

• 16² + 16² + 16² = 768.

768 is the sum of eight consecutive prime numbers: 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 = 768.

768 is 1100000000 in BASE 2, 30000 in BASE 4, and 300 in BASE 16.

768 is also a palindrome in a few bases:

• 363 BASE 15; note that 3(225) + 6(15) + 3(1) = 768
• OO BASE 31 (O = 24 base 10); note that 24(31) + 24(1) = 768
• CC BASE 63 (C = 12 base 10); note that 12(63) + 12(1) = 768
• 66 BASE 127; note that 6(127) + 6(1) = 768
• 33 BASE 255; note that 3(255) + 3(1) = 768

What do all those BASES have in common? They are all one less than a power of 2.

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