A Multiplication Based Logic Puzzle

Posts tagged ‘Square root’

671 is the Magic Sum of an 11 x 11 Magic Square

6 – 7 + 1 = 0 so 671 is divisible by 11.

671 is the sum of the fifteen prime numbers from 17 to 73.

Because 61 is one of its factors, 671 is the hypotenuse of the Pythagorean triple 121-660-671. The greatest common factor of those three numbers practically jumps out at me. Does it do the same thing to you?

Best of all 671 is the magic sum of an 11 x 11 magic square. (That link from wikipedia helped me construct this square. I’ll give directions so you can do it, too!)

671 Magic Sum for 11 x 11 Magic Square

Notice how every row, column, and diagonal on the square sums to 671. The reason it is the magic sum is because the sum of all the numbers from 1 to 121 can be computed and then divided by 11 (the number of rows). Here is the equation:

  • 671 = 121 x 120/2/11

Because 11 is an odd number there are simple directions to complete the entire square:

The number 1 is located in the exact center of the top row.

Find the number 2 on the square. (It’s located on the bottom row just right of the exact center square.) Notice that the numbers 3, 4, 5, and 6 are on the same diagonal. If you imagine the diagonal wrapping around the square, you can continue to follow it for numbers 7, 8, 9, 10, and 11. We can’t put the number 12 along the same diagonal because the number 1 is already in that spot, so we put the 12 UNDER the 11 and begin working on a new diagonal.

Anytime a number already occupies a space on a diagonal, put the next number under the preceding number and continue making a new diagonal. When a diagonal reaches the edge of the square, imagine that edge is connected to the opposite edge and continue the diagonal from the opposite edge.

I found it to be the trickiest placing the numbers 67 and 68, but other than that it was rather easy to know where to put the numbers.

Notice that the difference between any smaller number and the larger number just below it is either 12 or 1.

If you have excel on your computer, click on 12 Factors 2015-11-02, select the magic square tab, and then you can make this 11 x 11 magic square yourself. As you type in numbers, the columns, rows, and diagonals will automatically keep a running sum.

Once you get the square to give the magic sum in each direction, you can try doing the same thing with the 13 x 13 magic square that I’ve included on the same page. Its magic sum is 1105 which can be also be computed:

  • 1105 = 169 x 170/22/13.

There is actually many more possible and probably more complicated 11 x 11 and 13 x 13 magic squares. I hope you enjoy making some with this easy method.

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  • 671 is a composite number.
  • Prime factorization: 671 = 11 x 61
  • 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 671 has exactly 4 factors.
  • Factors of 671: 1, 11, 61, 671
  • Factor pairs: 671 = 1 x 671 or 11 x 61
  • 671 has no square factors that allow its square root to be simplified. √671 ≈ 25.903668.

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564 and Level 6

Consecutive primes 281 and 283 add up to 564.

564 is made from three consecutive numbers so it can be evenly divided by 3. If the middle number is divisible by 3, then a number made from three consecutive numbers will also be divisible by 9. Is 564 divisible by 9? Why or why not?

564 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 564 is a composite number.
  • Prime factorization: 564 = 2 x 2 x 3 x 47, which can be written 564 = (2^2) x 3 x 47
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 564 has exactly 12 factors.
  • Factors of 564: 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564
  • Factor pairs: 564 = 1 x 564, 2 x 282, 3 x 188, 4 x 141, 6 x 94, or 12 x 47
  • Taking the factor pair with the largest square number factor, we get √564 = (√4)(√141) = 2√141 ≈ 23.74868

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564 Logic

563 and Level 5

563 is prime, and it is the sum of all the prime numbers from 5 to 67.

563 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 563 is a prime number.
  • Prime factorization: 563 is prime and cannot be factored.
  • The exponent of prime number 563 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 563 has exactly 2 factors.
  • Factors of 563: 1, 563
  • Factor pairs: 563 = 1 x 563
  • 563 has no square factors that allow its square root to be simplified. √563 ≈ 23.727621

How do we know that 563 is a prime number? If 563 were not a prime number, then it would be divisible by at least one prime number less than or equal to √563 ≈ 23.7. Since 563 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 563 is a prime number.

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563 Logic

562 and Level 4

562 is the sum of all the prime numbers from 23 to 71.

562 is also the hypotenuse of the Pythagorean triple 320-462-562. Which factor of 562 is the greatest common factor of those three numbers?

562 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 562 is a composite number.
  • Prime factorization: 562 = 2 x 281
  • 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 562 has exactly 4 factors.
  • Factors of 562: 1, 2, 281, 562
  • Factor pairs: 562 = 1 x 562 or 2 x 281
  • 562 has no square factors that allow its square root to be simplified. √562 ≈ 23.7065.

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562 Logic

561 gives a false positive to these 102 prime number tests

We can use some easy divisibility tests to find two of the factors of 561.

  • 5 + 6 + 1 = 12, a multiple of 3 so 561 is divisible by 3.
  • 5 – 6 + 1 = 0, which is divisible by 11 so 561 can be evenly divided by 11.

But if we only do those divisibility tests, we will miss something very significant about the number 561:

If you divide 2^561 by 561, the remainder will be 2. When we are more interested in the remainder than the quotient, we can simply type “2, x^y, 561, Mod, 561, =” into the computer’s scientific calculator:

561 Mod Calculator

This is only a picture of a calculator.

2^561 (mod 561) = 2 means that 561 is VERY LIKELY a prime number, but this is one time when VERY LIKELY does not mean ACTUALLY!

561 has something in common with the number 341. Yes, both of them pass this quick prime number test, and both of them are composite numbers divisible by 11. Both numbers are called pseudo-prime numbers. (341 and 561 are the two smallest composite numbers to give a false positive to this particular test.)

561 is even more remarkable than 341:

  • 2^561 (mod 561) = 2, and 2^341 (mod 341) = 2 (Both numbers pass.)
  • 3^561 (mod 561) = 3, while 3^341 (mod 341) = 168 (561 passes; 341 fails.)
  • 5^561 (mod 561) = 5
  • 7^561 (mod 561) = 7
  • 11^561 (mod 561) = 11
  • 13^561 (mod 561) = 13
  • 17^561 (mod 561) = 17
  • etc.

There are 102 prime numbers less than 561, and p^561 (mod 561) = p for every single one of them! 561 acts like a prime number in those 102 ways.

In 1910 R. D. Carmichael discovered that 561 is the first COMPOSITE number that passes ALL those modular (remainder) prime number tests, so 561 is the first Carmichael number. Yes, there will be more – in fact, infinitely more.

R. D. Carmichael actually found that 561 passes ALL 559 prime number tests using each whole number between 1 and 561, for example 33^561 (mod 561) = 33. All prime numbers can make a similar claim, but 561 is the smallest composite number with that property.

(Note: I did not use the standard mathematical notation for this property, but what I used is equivalent to it and doesn’t require parenthesis when typing it into the computer’s scientific calculator. Also I think “=” is less intimidating looking for some of my readers than “≡”.)

There are other reasons why the number 561 is an interesting number:

Because 33 x 34/2 = 561, we know that 561 is the 33rd triangular number and is equal to 1 + 2 + 3 + . . . + 31 + 32 + 33, the sum of the first 33 whole numbers.

Because 17 x (2 x 17 – 1) = 561, we know that 561 is the 17th hexagonal number. (All hexagonal numbers are also triangular numbers.)

561 is also the hypotenuse of the Pythagorean triple 264-495-561. What is the greatest common factor of those three numbers? Hint: it is one of the factors of 561 listed below:

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  • 561 is a composite number.
  • Prime factorization: 561 = 3 x 11 x 17
  • 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 561 has exactly 8 factors.
  • Factors of 561: 1, 3, 11, 17, 33, 51, 187, 561
  • Factor pairs: 561 = 1 x 561, 3 x 187, 11 x 51, or 17 x 33
  • 561 has no square factors that allow its square root to be simplified. √561 ≈ 23.6854.

560 and Level 3

560 is the hypotenuse of the Pythagorean triple 336-448-560. What is the greatest common factor of those three numbers?

560 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 560 is a composite number.
  • Prime factorization: 560 = 2 x 2 x 2 x 2 x 5 x 7, which can be written 560 = (2^4) x 5 x 7
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 560 has exactly 20 factors.
  • Factors of 560: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560
  • Factor pairs: 560 = 1 x 560, 2 x 280, 4 x 140, 5 x 112, 7 x 80, 8 x 70, 10 x 56, 14 x 40, 16 x 35 or 20 x 28
  • Taking the factor pair with the largest square number factor, we get √560 = (√16)(√35) = 4√35 ≈ 23.664319

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. 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.

560 Factors

 

559 and Level 2

All the prime numbers from 67 to 97 add up to 559, and all the prime numbers from 103 to 127 also add up to 559.

559 is also the hypotenuse of Pythagorean triple 215-516-559. What is the greatest common factor of those three numbers?

559 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 559 is a composite number.
  • Prime factorization: 559 = 13 x 43
  • 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 559 has exactly 4 factors.
  • Factors of 559: 1, 13, 43, 559
  • Factor pairs: 559 = 1 x 559 or 13 x 43
  • 559 has no square factors that allow its square root to be simplified. √559 ≈ 23.64318

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559 Factors

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