764 and Level 4

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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.

764-factor-pairs

Here’s today’s puzzle:

764 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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764 Logic

763 and Level 3

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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.

763-factor-pairs

Now try solving today’s puzzle:

763 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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763 Factors

762 and Level 2

/ ivasallay / Leave a comment
  • 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.

762-factor-pairs

This level 2 puzzle isn’t very difficult:

 

762 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 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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762 Factors

761 and Level 1

/ ivasallay / Leave a comment
  • 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.

761-factor-pairs

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:

761 Puzzle

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.

761 Centered Square Number

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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761 Factors

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

/ ivasallay / Leave a comment
  • 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.

760-factor-pairs

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?

760 Puzzle

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:

760 Puzzle only

759 and Level 6

/ ivasallay / Leave a comment
  • 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.

759-factor-pairs

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

759 Puzzle

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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759 Logic

758 and Level 5

/ ivasallay / Leave a comment
  • 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.

758-factor-pairs

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.

758 Puzzle

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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758 Logic

 

 

757 and Level 4

/ ivasallay / 1 Comment
  • 757 is a prime number.
  • Prime factorization: 757 is prime.
  • The exponent of prime number 757 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 757 has exactly 2 factors.
  • Factors of 757: 1, 757
  • Factor pairs: 757 = 1 x 757
  • 757 has no square factors that allow its square root to be simplified. √757 ≈ 27.51363.

757-factor-pairs

How do we know that 757 is a prime number? If 757 were not a prime number, then it would be divisible by at least one prime number less than or equal to √757 ≈ 27.5. Since 757 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 757 is a prime number.

Here’s today’s puzzle. A logical way to solve it can be found in a table at the end of the post.

757 Puzzle

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

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Here’s another way we know that 757 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 9² = 757 with 26 and 9 having no common prime factors, 757 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √757 ≈ 27.5. Since 757 is not divisible by 5, 13, or 17, we know that 757 is a prime number.

757 is prime so the only way it can be written as the sum of consecutive numbers is 378 + 379 = 757.

757 is the sum of seven consecutive prime numbers: 97 + 101 + 103 + 107 + 109 + 113 + 127 = 757.

From OEIS.org we learn this distinguishing fact about the number 757:

757 reciprocal

Here are some square facts about the number 757:

26² + 9² = 757, and 468² + 595² = 757². That primitive Pythagorean triple, 468-595-757, can be calculated from 2(26)(9), 26² – 9², 26² + 9².

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

  • 24² + 10² + 9² = 757
  • 18² + 17² + 12² = 757

Finally 757 is a palindrome in four different bases:

  • 1001001 BASE 3; note that 1(3^6) + 0(3^5) + 0(3^4) + 1(3^3) + 0(3^2) + 0(3^1) + 1(3^0) = 757.
  • 757 BASE 10; note that 7(100) + 5(10) + 7(1) = 757.
  • 1F1 BASE 21 (F is 15 base 10); note that 1(21²) + 15(21) + 1(1) = 757.
  • 111 BASE 27; note that 1(27²) + 1(27) + 1(1) = 757.

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757 Logic

756 and Level 3

/ ivasallay / Leave a comment

Today’s Puzzle:

756 Puzzle

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

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 12.)  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.

756 Factors

Factor Trees for 756:

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Factors of 756:

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = 2² x 3³ x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

Sum-Difference Puzzles:

84 has six factor pairs. One of those pairs adds up to 25, and another one subtracts to 25. Put the factors in the appropriate boxes in the first puzzle.

756 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

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

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 756; that’s 24 consecutive numbers.
  • 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 + 40 + 41  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

 

755 and Level 2

/ ivasallay / Leave a comment
  • 755 is a composite number.
  • Prime factorization: 755 = 5 x 151
  • 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 755 has exactly 4 factors.
  • Factors of 755: 1, 5, 151, 755
  • Factor pairs: 755 = 1 x 755 or 5 x 151
  • 755 has no square factors that allow its square root to be simplified. √755 ≈ 27.4772633.

755-factor-pairs

Here’s today’s puzzle:

 

755 Puzzle

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

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Here’s a little more about 755:

755 is the sum of consecutive numbers three different ways:

  • 377 + 378 = 755; that’s 2 consecutive numbers.
  • 149 + 150 + 151 + 152 + 153 = 755; that’s 5 consecutive numbers.
  • 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 = 755; that’s 10 consecutive numbers.

Because 5 is one of its factors, 755 is the hypotenuse of Pythagorean triple 453-604-755.

755 is the sum of three squares six different ways:

  • 27² + 5² + 1² = 755
  • 25² + 11² + 3² = 755
  • 25² + 9² + 7² = 755
  • 23² + 15² + 1² = 755
  • 21² + 17² + 5² = 755
  • 19² + 15² + 13² = 755

755 is palindrome 131 in BASE 26; note that 1(26²) + 3(26) + 1(1) = 755.

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755 Factors