• 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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• 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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Here’s some more thoughts I had about the number 764:

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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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. OEIS.org informs us 763 is the smallest number whose 4th power can make that claim.

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• 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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