A Multiplication Based Logic Puzzle

Archive for the ‘Level 4 Puzzle’ Category

791 and Level 4

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

  • 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
  • 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

791-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 791 is a composite number.
  • Prime factorization: 791 = 7 x 113
  • 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 791 has exactly 4 factors.
  • Factors of 791: 1, 7, 113, 791
  • Factor pairs: 791 = 1 x 791 or 7 x 113
  • 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791-factor-pairs

785 and Level 4

  • 785 is a composite number.
  • Prime factorization: 785 = 5 x 157
  • 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 785 has exactly 4 factors.
  • Factors of 785: 1, 5, 157, 785
  • Factor pairs: 785 = 1 x 785 or 5 x 157
  • 785 has no square factors that allow its square root to be simplified. √785 ≈ 28.01785.

785-factor-pairs

Now for today’s puzzle:

785-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 785:

785 is the sum of two squares two different ways:

  • 28² + 1² = 785
  • 23² + 16² = 785

785 is also the sum of three squares three different ways:

  • 26² + 10² + 3² = 785
  • 25² + 12² + 4² = 785
  • 19² + 18² + 10² = 785

Because its prime factorization is 5 × 157 (two numbers that are also the sum of two squares), 785 is the hypotenuse of four Pythagorean triples, two of which are primitive triples:

  • 56-783-785 primitive calculated from 2(28)(1), 28² – 1², 28² + 1²
  • 273-736-785 primitive calculated from 23² – 16², 2(23)(16), 23² + 16²
  • 425-660-785 which is 5 times 85-132-157
  • 471-628-785 which is 157 times 3-4-5

785 is also a palindrome in two different bases:

  • 555 BASE 12; note that 5(144) + 5(12) + 5(1) = 785
  • 101 BASE 28; note that 1(28²) + 0(28) + 1(1) = 785

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785-logic

 

778 and Level 4

  • 778 is a composite number.
  • Prime factorization: 778 = 2 x 389
  • 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 778 has exactly 4 factors.
  • Factors of 778: 1, 2, 389, 778
  • Factor pairs: 778 = 1 x 778 or 2 x 389
  • 778 has no square factors that allow its square root to be simplified. √778 ≈ 27.892651.

778-factor-pairs

Here’s today’s puzzle. Hints for solving it are in a table at the end of the post.

 

778 Puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Stetson.edu informs us that 778 is the number of ways  a 1 x 5 rectangle can be surrounded by other 1 x 5 rectangles. I’m not going to publish or even find all 778 possible ways, but here is one of them:

778 Surround

27² + 7² = 778.

We can use those numbers to compute a Pythagorean triple: 2(27)(7), 27² – 7², 27² + 7², which turns out to be 378-680-778, and thus 378² + 680² = 778².

778 is also the sum of three squares three different ways:

  • 25² + 12² + 3² = 778
  • 24² + 11² + 9² = 778
  • 21² + 16² + 9² = 778

778 is palindrome 1G1 BASE 21 (G is 16 base 10)

That palindrome means that 1(21²) + 16(21) + 1(1) = 778.

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778 Logic

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.

771-factor-pairs

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:

771 Puzzle

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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771 Logic


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.

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

757 and Level 4

  • 757 is a prime number.
  • Prime factorization: 757 is prime and cannot be factored.
  • 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 Stetson.edu 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

750 and Level 4

  • 750 is a composite number.
  • Prime factorization: 750 = 2 x 3 x 5 x 5 x 5, which can be written 750 = 2 x 3 x (5^3)
  • The exponents in the prime factorization are 1, 1, and 3. Adding one to each and multiplying we get (1 + 1)(1 + 1)(3 + 1) = 2 x 2 x 4 = 16. Therefore 750 has exactly 16 factors.
  • Factors of 750: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750
  • Factor pairs: 750 = 1 x 750, 2 x 375, 3 x 250, 5 x 150, 6 x 125, 10 x 75, 15 x 50, or 25 x 30
  • Taking the factor pair with the largest square number factor, we get √750 = (√25)(√30) = 5√30 ≈ 27.386127875.

750-factor-pairs

Here’s today’s puzzle. Some steps to solving it is given in the table at the end of the post.

750 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

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Here’s more reasons to be interested in the number 750:

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

  • 249 + 250 + 251 = 750; that’s 3 consecutive numbers.
  • 186 + 187 + 188 + 189 = 750; that’s 4 consecutive numbers.
  • 148 + 149 + 150 + 151 + 152 = 750; that’s 5 consecutive numbers.
  • 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 = 750; that’s 12 consecutive numbers.
  • 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 = 750; that’s 15 consecutive numbers.
  • 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 750; that’s 20 consecutive numbers.
  • 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 750; that’s 25 consecutive numbers.

750 is also the sum of all the prime numbers from 17 to 79. Do you remember what those sixteen prime numbers are?

Because 5, 25, and 125 are its factors, 750 is the hypotenuse of three Pythagorean triple triangles:

  • 210² + 720² = 750²
  • 264² + 702² = 750²
  • 450² + 600² = 750²

750² = 562500 which is another cool looking square number whose digits include 5 and the value of 5^4.

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

  • 26² + 7² + 5² = 750
  • 25² + 11² + 2² = 750
  • 25² + 10² + 5² = 750
  • 23² + 14² + 5² = 750
  • 23² + 11² + 10² = 750
  • 19² + 17² + 10² = 750

Wikipedia tells us that 750 is the 15th nonagonal number because 15(7⋅15 – 5)/2 = 750. It is also 10 times the 5th nonogonal number because 10 ⋅ 5(7⋅5 – 5)/2 = 750.

750 is a palindrome in three different bases:

  • 23232 BASE 4; note that 2(4^4) + 3(4^3) + 2(4^2) + 3(4^1) + 2(4^0) = 750.
  • 2A2 BASE 17 (A= 10 base 10); note that 2(17²) + 10(17) + 2(1) = 750.
  • PP BASE 29 (P = 25 base 10); note that 25(29) + 25(1) = 750.

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750 Logic

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