A Multiplication Based Logic Puzzle

Archive for the ‘Level 2 Puzzle’ Category

796 and a Valentine

Here’s a puzzle you can print, cut out, and give as a valentine:

796-puzzle

You may know the divisibility rules for these powers of two:

  • If the last digit of a number is divisible by 2, the whole number is divisible by 2.
  • If the last two digits are divisible by 4, the whole number is divisible by 4.
  • If the last three digits are divisible by 8, the whole number is divisible by 8.

But I’m going to apply some other time-saving but possibly more confusing divisibility rules to the number 796:

  • 796 is divisible by 2 because 6 is an even number.
  • 796 is divisible by 4 because even number 6 is NOT divisible by 4, and 9 is an odd number.
  • 796 is NOT divisible by 8 because 96 is divisible by 8, and 7 is an odd number.

Because 796 is divisible by 4 but not by 8, it can be written as the sum of 8 consecutive numbers:

  • 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 = 796

796 is also the sum of all the prime numbers from 113 to 149:

  • 113 + 127 + 131 + 137 + 139 + 149 = 796

Here is the factoring information for 796:

  • 796 is a composite number.
  • Prime factorization: 796 = 2 x 2 x 199, which can be written 796 = (2^2) x 199
  • 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 796 has exactly 6 factors.
  • Factors of 796: 1, 2, 4, 199, 398, 796
  • Factor pairs: 796 = 1 x 796, 2 x 398, or 4 x 199
  • Taking the factor pair with the largest square number factor, we get √796 = (√4)(√199) = 2√199 ≈ 28.21347.

796-factor-pairs

 

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789 and Level 2

789 consists of exactly three consecutive numbers so it is divisible by 3.

789-puzzle

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

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  • 789 is a composite number.
  • Prime factorization: 789 = 3 x 263
  • 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 789 has exactly 4 factors.
  • Factors of 789: 1, 3, 263, 789
  • Factor pairs: 789 = 1 x 789 or 3 x 263
  • 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789-factor-pairs

783 and Level 2

  • 783 is a composite number.
  • Prime factorization: 783 = 3 x 3 x 3 x 29, which can be written 783 = (3^3) x 29
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 783 has exactly 8 factors.
  • Factors of 783: 1, 3, 9, 27, 29, 87, 261, 783
  • Factor pairs: 783 = 1 x 783, 3 x 261, 9 x 87, or 27 x 29
  • Taking the factor pair with the largest square number factor, we get √783 = (√9)(√87) = 3√87 ≈ 27.982137.

783-factor-pairs

Here’s today’s puzzle. It’s a level 2 so it isn’t very difficult:

783-puzzle

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

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27 x 29 = 783. Since (n – 1)(n + 1) always equals n² – 1, we know that 783 is one number away from the next perfect square.

29 is a factor of 783, making 783 the hypotenuse of a Pythagorean triple:

  • 540-567-783, which is 27 times 20-21-29.
  • Thus 540² + 567² = 783² just as 20² + 21² = 29².

783 is also a palindrome in bases 15, 23, and 28:

  • 373 BASE 15; note that 3(225) + 7(15) + 3(1) = 783
  • 1B1 BASE 23 (B is 11 base 10); note that 1(23²) + 11(23) + 1(1) = 783
  • RR BASE 28 (R is 27 base 10); note that 27(28) + 27 = 783

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783-factors

775 and Level 2

  • 775 is a composite number.
  • Prime factorization: 775 = 5 x 5 x 31, which can be written 775 = (5^2) x 31
  • 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 775 has exactly 6 factors.
  • Factors of 775: 1, 5, 25, 31, 155, 775
  • Factor pairs: 775 = 1 x 775, 5 x 155, or 25 x 31
  • Taking the factor pair with the largest square number factor, we get √775 = (√25)(√31) = 5√31 ≈ 27.83882181.

775-factor-pairs

Here’s today’s factoring puzzle:

775 Puzzle

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

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Here is more about the number 775:

775 is part of several Pythagorean triples including two that are primitive:

  • 168-775-793 (Primitive)
  • 775-1860-2015
  • 775-9672-9703
  • 775-12000-12025
  • 775-300312-300313 (Primitive)

775 is palindrome PP in BASE 30 (P = 25 base 10). Note that 25(30) +25(1) = 775.

775 is also the sum of three triangular numbers 9 different ways:

  1. 3+276+ 496
  2. 6+28+741
  3. 6+66+703
  4. 10+300+465
  5. 15-325-435
  6. 36+36+703
  7. 78+136+561
  8. 171+253+351
  9. 120+190+465

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775 Factors

 


769 and Level 2

  • 769 is a prime number.
  • Prime factorization: 769 is prime and cannot be factored.
  • 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.

769-factor-pairs

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:

 

769 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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769 Factors


762 and Level 2

  • 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

755 and Level 2

  • 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

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