• 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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• 773 is a prime number.
• Prime factorization: 773 is prime.
• The exponent of prime number 773 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 773 has exactly 2 factors.
• Factors of 773: 1, 773
• Factor pairs: 773 = 1 x 773
• 773 has no square factors that allow its square root to be simplified. √773 ≈ 27.8028775.

How do we know that 773 is a prime number? If 773 were not a prime number, then it would be divisible by at least one prime number less than or equal to √773 ≈ 27.8. Since 773 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 773 is a prime number.

Here is today’s puzzle for you to try to solve:

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

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What else is special about the number 773?

22² + 17² = 773 so 773 is the hypotenuse of the primitive Pythagorean triple 195-748-773 which was calculated using 22² – 17², 2(17)(22), 22² + 17².

Thus 195² + 748² + 773².

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

• 26² + 9² + 4² = 773
• 25² + 12² + 2² = 773
• 24² + 14² + 1² = 773
• 23² + 12² + 10² = 773
• 22² + 15² + 8² = 773
• 20² + 18² + 7² = 773

773 is a palindrome in two other bases:

• 545 BASE 12, note that 5(144) + 4(12) + 5(1) = 773
• 3D3 BASE 14 (D = 13 base 10); note that 3(196) + 13(14) + 3(1) = 773

Here’s another way we know that 773 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 22² + 17² = 773 with 22 and 17 having no common prime factors, 773 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √773 ≈ 27.8. Since 773 is not divisible by 5, 13, or 17, we know that 773 is a prime number.

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• 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.

OEIS.org 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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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 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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Here is more information about the number 770:

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

• 768 is a composite number.
• Prime factorization: 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3, which can be written 768 = 2⁸ × 3
• The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 × 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 × 768, 2 × 384, 3 × 256, 4 × 192, 6 × 128, 8 × 96, 12 × 64, 16 × 48 or 24 × 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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