525 = (23 + 2)(23 – 2) = (23^2) – (2^2) makes 525 the longer leg in what primitive Pythagorean triple?

Why do people enjoy number puzzles? A mathemagician friend once stated that a key reason was loving to solve mysteries.

I find making and solving puzzles to be quite relaxing. Solve the mystery or relax a little finding the factors of this puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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• 525 is a composite number.
• Prime factorization: 525 = 3 x 5 x 5 x 7, which can be written 525 = 2 x (5^2) x 7
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 525 has exactly 12 factors.
• Factors of 525: 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525
• Factor pairs: 525 = 1 x 525, 3 x 175, 5 x 105, 7 x 75, 15 x 35, or 21 x 25
• Taking the factor pair with the largest square number factor, we get √525 = (√25)(√21) = 5√21 ≈ 22.912878

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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.

24 is divisible by 4 and that means 524 is also divisible by 4.

Numbers that are divisible by 4 can have their square roots reduced. 524 ÷ 4 = 131, a prime number whose only square factor is 1, so √524 = (√4)(√131) = 2√131.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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• 524 is a composite number.
• Prime factorization: 524 = 2 x 2 x 131, which can be written 524 = (2^2) x 131
• 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 524 has exactly 6 factors.
• Factors of 524: 1, 2, 4, 131, 262, 524
• Factor pairs: 524 = 1 x 524, 2 x 262, or 4 x 131
• Taking the factor pair with the largest square number factor, we get √524 = (√4)(√131) = 2√131 ≈ 22.891046

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523 = 61 + 67 + 71 + 73 + 79 + 83 + 89 which are all the prime numbers between 60 and 96.

521 and 523 are twin primes.

523 is the 99th prime number. It will take longer than ever before for another prime number to be featured on this blog.

The prime gap is defined as the difference between two consecutive prime numbers. The gap between this prime number and the next one is greater than between any two previous prime numbers.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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• 523 is a prime number.
• Prime factorization: 523 is prime.
• The exponent of prime number 523 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 523 has exactly 2 factors.
• Factors of 523: 1, 523
• Factor pairs: 523 = 1 x 523
• 523 has no square factors that allow its square root to be simplified. √523 ≈ 22.869193

How do we know that 523 is a prime number? If 523 were not a prime number, then it would be divisible by at least one prime number less than or equal to √523 ≈ 22.9. Since 523 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 523 is a prime number.

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521 = 20² + 11², and 521  is the hypotenuse of the primitive Pythagorean triple 279-440-521.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 521 is a prime number.
• Prime factorization: 521 is prime.
• The exponent of prime number 521 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 521 has exactly 2 factors.
• Factors of 521: 1, 521
• Factor pairs: 521 = 1 x 521
• 521 has no square factors that allow its square root to be simplified. √521 ≈ 22.8254244

How do we know that 521 is a prime number? If 521 were not a prime number, then it would be divisible by at least one prime number less than or equal to √521 ≈ 22.8. Since 521 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 521 is a prime number.

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Finding something interesting about the number 518 was as easy as 1-2-3.

518 = (5^1) + (1^2) + (8^3). Thank you OEIS.org for that fun fact.

518 is also the hypotenuse of the Pythagorean triple 168-490-518. Can you find the greatest common factors of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 518 is a composite number.
• Prime factorization: 518 = 2 x 7 x 37
• 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 518 has exactly 8 factors.
• Factors of 518: 1, 2, 7, 14, 37, 74, 259, 518
• Factor pairs: 518 = 1 x 518, 2 x 259, 7 x 74, or 14 x 37
• 518 has no square factors that allow its square root to be simplified. √518 ≈ 22.759613

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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.

5 – 1 + 7 = 11 so 517 can be evenly divided by 11.

517 = 97 + 101 + 103 + 107 + 109 which is all the prime numbers between 90 and 112.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 517 is a composite number.
• Prime factorization: 517 = 11 x 47
• 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 517 has exactly 4 factors.
• Factors of 517: 1, 11, 47, 517
• Factor pairs: 517 = 1 x 517 or 11 x 47
• 517 has no square factors that allow its square root to be simplified. √517 ≈ 22.737634

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Look at this chart of the sum of the factors for the numbers 1 – 25:

If we made the chart infinitely long using every counting number as n, there are certain numbers like 2, 5, 52, 88, and 96 that will NEVER appear in either column C or column D. Those numbers are called untouchable numbers, and 516 is one of them. Even though there are relatively few untouchable numbers, Paul Erdős proved that there are infinitely many of them.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 516 is a composite number.
• Prime factorization: 516 = 2 x 2 x 3 x 43, which can be written 516 = (2^2) x 3 x 43
• 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 516 has exactly 12 factors.
• Factors of 516: 1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516
• Factor pairs: 516 = 1 x 516, 2 x 258, 3 x 172, 4 x 129, 6 x 86, or 12 x 43
• Taking the factor pair with the largest square number factor, we get √516 = (√4)(√129) = 2√129 ≈ 22.715633382

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