A Multiplication Based Logic Puzzle

Archive for the ‘Level 2 Puzzle’ Category

803 From Top to Bottom

8 – 0 + 3 = 11, so 803 is divisible by 11.

  • 803 is a composite number.
  • Prime factorization: 803 = 11 x 73
  • 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 803 has exactly 4 factors.
  • Factors of 803: 1, 11, 73, 803
  • Factor pairs: 803 = 1 x 803 or 11 x 73
  • 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

  • 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

  • 27² + 7² + 5² = 803
  • 25² + 13² + 3² = 803
  • 23² + 15² + 7² = 803
  • 21² + 19² + 1² = 803
  • 19² + 19² + 9² = 803
  • 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

  • 263 + 269 + 271 = 803, that’s 3 consecutive primes.
  • 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
  • 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

  • 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
  • 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

 

 

 

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802 Pi Day at Smith’s

In the United States tomorrow’s date is written 3-14. Because 3.14 is a famous approximation for π (pi), people all over the country will eat pie to celebrate Pi Day. This afternoon I took a picture of this sign and the pie display at my local Smith’s Food and Drug.

I took that picture right when I walked into the store, but there were no pies on display for National Pi Day.

About 15 minutes later I returned to the display to take another picture. Now there were pies on the table! I told a salesperson who I think worked on the display that I was going to take a picture and put it on my blog. She asked what kind of a blog I wrote. I told her a math blog. She looked puzzled and asked why I would want to put a picture of pies on a math blog. Then she turned around, looked at the display, and said something like, “Oh, now I get it, the number pi.”

How do you choose between apple, cherry, or peach pie? It’s much easier if you choose two and then you can get a free 8 oz. Cool Whip, too. Yummy.

If by chance you prefer pizza pi, here’s a thought from twitter that is often repeated in March:

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And here’s some original artwork that displays pi in a way I had never thought of before:

//platform.twitter.com/widgets.js

BREAKING: secret of Pi revealed #PiDay pic.twitter.com/Ao8BQp31jd

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You can also look here for a million digits of pi.

But pi is not the only interesting number in the world. Every number has its own curiosities. Let me tell you some reasons to get excited about the number 802:

802 is the sum of two squares:

  • 21² + 19² = 802

So 802 is the hypotenuse of a Pythagorean triple:

  • 80-798-802, which is 2 times another triple: 40-399-401.

It also means something else: Since odd numbers 21 and 19 have no common prime factors, 802 can be evenly divide by 2. Duh. . ., but it also means that unless 802 is also divisible by 5, 13, or 17, its only factors will be 2 and a prime number! Why are those three numbers the only ones I care about? Because they are the only prime number Pythagorean triple hypotenuses less than √802 ≈ 28.3.

Guess what? 5, 13, and 17 do not divide evenly into 802, so 802 is the product of 2 and a prime number which happens to be 401.

  • 802 is a composite number.
  • Prime factorization: 802 = 2 x 401
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 802 has exactly 4 factors.
  • Factors of 802: 1, 2, 401, 802
  • Factor pairs: 802 = 1 x 802 or 2 x 401
  • 802 has no square factors that allow its square root to be simplified. √802 ≈ 28.3196045

Today’s puzzle is number 802 to distinguish it from every other puzzle I’ve made. Writing the numbers 1 – 10 in both the top row and the first column so that the factors and the clues work together as a multiplication table is as easy as pie!

Print the puzzles or type the solution on this excel file: 10-factors 801-806

And here is a little more about the number 802:

802 is the sum of 8 consecutive prime numbers:

  • 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 802

802 can also be written as the sum of three squares three different ways:

  • 28² + 3² + 3² = 802
  • 27² + 8² + 3² = 802
  • 24² + 15² + 1² = 802

802 is also a palindrome in two other bases:

  • 414 BASE 14 because 4(196) + 1(14) + 4(1) = 802
  • 202 BASE 20 because 2(400) + 0(20) + 2(1) = 802

 

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

 

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

789 is the sum of consecutive prime numbers 2 different ways:

  • 257 + 263 + 269 = 789 (that’s 3 consecutive primes.)
  • 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 789 (that’s 21 consecutive primes!)

789 is the sum of three squares eight different ways:

  • 28² + 2² + 1² = 789
  • 26² + 8² + 7² = 789
  • 25² + 10² + 8² = 789
  • 23² + 16² + 2² = 789
  • 23² + 14² + 8² = 789
  • 22² + 17² + 4² = 789
  • 22² + 16² + 7² = 789
  • 20² + 17² + 10² = 789

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


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